THE   CALCULUS 


A  SERIES  OF  MATHEMATICAL  TEXTS 

EDITED    BY 

EARLE  RAYMOND   HEDRICK 

Brenke. 
PLANE   AND  SOLID  ANALYTIC   GEOMETRY 

""b, A..-™- ^-" -^ ^-" ^^"='  ::;„TH 

PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
COMPLETE   TABLES 
B,  A„K..  Mo»«o.  K.».o.  and  Lo.-s  I»»o.p. 
PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
BRIEF   TABLES 
By  A.rKED  Mo.koe  Kenvon  and  Louis  I.oo.d. 

THE   MACMILLAN   TABLES  ,,  havmond  Hedrick. 

Prepared  under  the  direction  of  Earle  Ra.mod 

^^Tw:r:™i»  .o..  a.,  c„.....  a.,»...u». 


THE    CALCULUS 


BY 

ELLERY   WILLIAMS   DAVIS 

H 
PKOFE880K   OF    MATHEMATICS,    THE    UNIVERSITY    OF    NEBRASKA 

ASSISTED    BY 

WILLIAM  CHARLES   BRENKE 

ASSOCIATE    PROFESSOR    OF    MATHEMATICS,    THE    UNIVERSITY    OF    NEBRASKA 


EDITED    BY 

EARLE   RAYMOND  HEDRICK 


THE   MACMILLAN   COMPANY 
1915 

All  rights  reserved 


Copyright,  1912, 
By  the   MACMILLAN  COMPANY.  , 

Set  up  and  electrotyped.     Published  September,  1912.     Reprinted 
October,  191a ;   May,  iqt^;  August,   October,  1914;   February,  1915.  tti 


lilf-  ^     V^-u—  U-wt-^       ^M-^4  Ct^ 


Nortoooli  ^tf«8 

J.  8.  Gushing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

The  significance  of  the  Calculus,  the  possibility  of  applying 
it  in  other  fields,  its  usefulness,  ought  to  be  kept  constantly 
ind  vividly  before  the  student  during  his  study  of  the  subject, 
rather  than  be  deferred  to  an  uncertain  future. 

Not  only  for  students  who  intend  to  become  engineers,  but 
ilso  for  those  planning  a  profound  study  of  other  sciences,  the 
isef ulness  of  the  Calculus  is  universally  recognized  by  teachers ; 
t  should  be  consciously  realized  by  the  student  himself.  It  is 
)bvious  that  students  interested  primarily  in  mathematics, 
particularly  if  they  expect  to  instruct  others,  should  recognize 
;he  same  fact. 

To  all  these,  and  even  to  the  student  who  expects  only  gen- 
jral  culture,  the  use  of  certain  types  of  applications  tends  to 
nake  the  subject  more  real  and  tangible,  and  offers  a  basis  for 
m  interest  that  is  not  artificial.  Such  an  interest  is  necessary 
;o  secure  proper  attention  and  to  insure  any  real  grasp  of  the 
essential  ideas. 

For  this  reason,  the  attempt  is  made  in  this  book  to  present 
IS  many  and  as  varied  applications  of  the  Calculus  as  it  is 
possible  to  do  without  venturing  into  technical  fields  whose 
lubject  matter  is  itself  unknown  and  incomprehensible  to  the 
;tudent,  and  Avithout  abandoning  an  orderly  presentation  of 
fundamental  principles. 

The  same  general  tendency  has  led  to  the  treatment  of 
lOpics  with  a  view  toward  bringing  out  their  essential  useful- 
less.  Thus  the  treatment  of  the  logarithmic  derivative  is 
vitalized  by  its  presentation  as  the  relative  rate  of  change  of  a 
quantity;  and  it  is  fundamentally  connected  with  the  important 
'  compound  interest  law,"  which  arises  in  any  phenomenon  in 


VI  PREFACE 

which  the  relative  rate  of  increase  (logarithmic  derivative)  is 
constant. 

Another  instance  of  the  same  tendency  is  the  attempt,  in  the 
introduction  of  the  precise  concept  of  curvature,  to  explain  the 
reason  for  the  adoption  of  this,  as  opposed  to  other  simpler 
but  cruder  measures  of  bending.  These  are  only  instances,  of 
two  typical  kinds,  of  the  way  in  which  the  effort  to  bring  out 
the  usefulness  of  the  subject  has  influenced  the  j)resentation  of 
even  the  traditional  topics. 

Rigorous  forms  of  demonstration  are  not  insisted  upon,  es- 
pecially where  the  precisely  rigorous  proofs  would  be  beyond 
the  present  grasp  of  the  student.  Rather  the  stress  is  laid  upon 
the  student's  certain  comprehension  of  that  which  is  done,  and 
his  conviction  that  the  results  obtained  are  both  reasonable  and 
useful.  At  the  same  time,  an  effort  has  been  made  to  avoid 
those  grosser  errors  and  actual  misstatements  of  fact  which 
have  often  offended  the  teacher  in  texts  otherwise  attractive 
and  teachable. 

Thus  a  proof  for  the  formula  for  differentiating  a  logarithm 
is  given  which  lays  stress  on  the  very  meaning  of  logarithms ; 
while  it  is  not  absolutely  rigorous,  it  is  at  least  just  as  rigorous 
as  the  more  traditional  proof  which  makes  use  of  the  limit  of 
(1  -I-  1/?*)"  as  n  becomes  infinite,  and  it  is  far  more  convincing 
and  instructive.  The  proof  used  for  the  derivative  of  the  sine 
of  an  angle  is  quite  as  sound  as  the  more  traditional  proof 
(which  is  also  indicated),  and  makes  use  of  fundamentally  use- 
ful concrete  concepts  connected  with  circular  motion.  These 
two  proofs  again  illustrate  the  tendency  to  make  the  subject 
vivid,  tangible,  and  convincing  to  the  student;  this  tendency 
will  be  found  to  dominate,  in  so  far  as  it  was  found  possible, 
every  phase  of  every  topic. 

Many  traditional  theorems  are  omitted  or  reduced  in  impor- 
tance. In  many  cases,  such  theorems  are  reproduced  in  exer- 
cises, with  a  sufficient  hint  to  enable  the  student  to  master 
them.     Thus  Taylor's  Theorem  in  several  variables,  for  which 


PREFACE  vil 

wide  applications  are  not  apparent  until  further  study  of  nuithe- 
niatics  and  science,  is  presented  in  this  manner. 

On  the  other  hand,  many  theorems  of  importance,  both  from 
mathematical  and  scientific  grounds,  which  have  been  omitted 
traditionally,  are  included.  Examples  of  this  sort  are  the  brief 
treatment  of  simple  harmonic  motion,  the  wide  application  of 
Cavalieri's  theorem  and  the  prismoid  formula,  other  api)roxi- 
mation  formulas,  the  theory  of  least  squares  (under  the  head 
of  exercises  in  maxima  and  minima),  and  many  other  topics. 

The  Exercises  throughout  are  colored  by  the  views  expressed 
above,  to  bring  out  the  usefulness  of  the  subject  and  to  give 
tangible  concrete  meaning  to  the  concepts  involved.  Yet  formal 
exercises  are  not  at  all  avoided,  nor  is  this  necessary  if  the 
student's  interest  has  been  secured  through  conviction  of  the 
usefulness  of  the  topics  considered.  Far  more  exercises  are 
stated  than  should  be  attempted  by  any  one  student.  This  will 
lend  variety,  and  will  make  possible  the  assignment  of  different 
problems  to  different  students  and  to  classes  in  successive 
years.  It  is  urged  that  care  be  taken  in  selecting  from  the 
exercises,  since  the  lists  are  graded  so  that  certain  groups  of 
exercises  prepare  the  student  for  other  groups  which  follow ; 
but  it  is  unnecessary  that  all  of  any  group  be  assigned,  and  it  is 
urged  that  in  general  less  than  half  be  used  for  any  one  stu- 
dent. Exercises  that  involve  practical  applications  and  others 
that  involve  bits  of  theory  to  be  worked  out  by  the  student  are 
of  frequent  occurrence.  These  should  not  be  avoided,  for  they 
are  in  tune  with  the  spirit  of  the  whole  book ;  great  care  has 
vbeen  taken  to  select  these  exercises  to  avoid  technical  concepts 
strange  to  the  student  or  proofs  that  are  too  difficult. 

An  effort  is  made  to  remove  many  technical  difficulties  by 
the  intelligent  use  of  tables.  Tables  of  Integrals  and  many 
other  useful  tables  are  appended ;  it  is  hoped  that  these  will 
be  found  usable  and  helpful. 

Parts  of  the  book  may  be  omitted  without  destroying  the 
essential  unity  of  the  whole.     Thus  the  rather  complete  treat- 


Viu  PREFACE 

ment  of  Differential  Equations  (of  the  more  elementary  types) 
can  be  omitted.  Even  the  chapter  on  Functions  of  Several 
Variables  can  be  omitted,  at  least  except  for  a  few  paragraphs, 
without  vital  harm ;  and  the  same  may  be  said  of  the  chapter 
on  Approximations.  The  omission  of  entire  chapters,  of  course, 
would  only  be  contemplated  where  the  pressure  of  time  is  un- 
usual ;  but  many  paragraphs  may  be  omitted  at  the  discretion 
of  the  teacher. 

Although  care  has  been  exercised  to  secure  a  consistent  order 
of  topics,  some  teachers  may  desire  to  alter  it ;  for  example, 
an  earlier  introduction  of  transcendental  functions  and  of  por- 
tions of  the  chapter  on  Approximations  may  be  desired,  and  is 
entirely  feasible.  But  it  is  urged  that  the  comparatively  early 
introduction  of  Integration  as  a  summation  process  be  retained, 
since  this  further  impresses  the  usefulness  of  the  subject,  and 
accustoms  the  student  to  the  ideas  of  derivative  and  integral 
before  his  attention  is  diverted  by  a  variety  of  formal  rules. 

Purely  destructive  criticism  and  abandonment  of  coherent 
arrangement  are  just  as  dangerous  as  ultra-conservatism.  This 
book  attempts  to  preserve  the  essential  features  of  the  Calculus, 
to  give  the  student  a  thorough  training  in  mathematical  rea- 
soning, to  create  in  him  a  sure  mathematical  imagination,  and 
to  meet  fairly  the  reasonable  demand  for  enlivening  and  en- 
riching the  subject  through  applications  at  the  expense  of  purely 
formal  work  that  contains  no  essential  principle. 

E.  W.DAVIS, 

W.  C.  BRENKE, 

E.  R.  HEDRICK,  Editor. 

June,  1912. 


CONTENTS 

f  Pajre  numbers  in  Roman  type  refer  to  the  body  of  the  book  ;  those  in  italic  type  refer  to 
pages  of  the  Tables.] 

PAGES 

CHAPTER   I        FUNCTIONS 1-6 

§  1.    Dependence 1 

§  2.    Variables.     Constants.     Functions        .        .        .        ,1-2 

Exercises  I.     Functions  and  Graphs        .....  2-3 

§  3.    The  Function  Notation 3 

Exercises  II.     Substitution.     Function  Notation     .        .         ,  3-5 

CHAPTER   II        RATES-^  LIMITS        DERIVATIVES        .  6-27 

§  4.    Rate  of  Increase.     Slope 6-8 

§  5.   General  Rules 8 

§  6.    Slope  Negative  or  Zero.     [Maxima  and  Minima.]          .  8-11 

Exercises  III.     Slopes  of  Curves 11-12 

§  7.    Speed 12-1-4 

§  8.    Component  Speeds 14 

§  9.   Continuous  Functions 14-15 

Exercises  IV.     Speed 15-16 

§  10.    Limits.     Infinitesimals 16-17 

§  11.    Properties  of  Limits 17-18 

§  12.    Ratio  of  an  Arc  to  its  Chord 18-19 

§  13.    Ratio  of  the  Sine  of  an  Angle  to  the  Angle    .        .         .19 

§  14.    Infinity 19-20 

Exercises  V.     Limits  and  Infinitesimals          ....  20-22 

§  15.    Derivatives 22-23 

§  16.    Formula  for  Derivatives 23-24 

§  17.    Rule  for  Differentiation 24-26 

Exercises  VI.     Formal  Differentiation 26-27 

CHAPTER    III  DIFFERENTIATION     OF     ALGEBRAIC 

FUNCTIONS 28-57 

Part  I         Explicit  Functions    .         .         .  28-43 

§  18.    Classification  of  Functions 28-29 

§  19.   Differentiation  of  Polynomials 29-31 

iz 


CONTENTS 


Exercises  VII.     Differentiation  of  Polynomials 

§  20.    Differentiation  of  Rational  Functions.     [Quotient.] 

Exercises  VIII.     Differentiation  of  Rational  Functions 

§  21.    Derivative  of  a  Product 

§  22.    Derivative  of  a  Function  of  a  Function 
Exercises  IX.     Short  Methods.     Rational  Functions 
§  23.   Differentiation  of  Irrational  Functions  . 

§  24.    Collection  of  Formulas 

§  25.    Illustrative  Examples  of  Irrational  Functions 
Exercises  X.     Algebraic  Functions 


PAGES 

31-32 
32-33 
34-35 
35-36 
36-37 
37-38 
38-40 
40-41 
41-42 
42-43 


Part  II        Equations  not  in  Explicit  Form        Differentials  44-57 

§  26.    Solution  of  Equations 44-45 

§  27.   Explicit  and  Implicit  Functions 45-46 

§  28.    Inverse  Functions 46-47 

§  29.    Parameter  Forms 47 

Exercises  XI.     Functions  not  in  Explicit  Form       .         .         .  47-49 

§  30.    Rates 49-50 

§  31.    The  Differential  Notation 50-52 

§  32.    Differential  Formulas 52-53 

Exercises  XII.     Differentials 54-57 


CHAPTER   IV        FIRST   APPLICATIONS   OF    DIFFEREN- 
TIATION        58-90 

Part  I        Application  to  Curves        Extremes      .  58-70 

§  33.    Tangents  and  Normals 58-59 

§  34.    Tangents  and  Normals  for  Curves  not  in  Explicit  Form  59 

§  35.    Secondary  Quantities 60 

§  36.    Illustrative  Examples 60-61 

Exercises  XIII.     Tangents  and  Normals  ....  62-63 

§  37.    Extremes.     [Maxima  and  Minima  ]      .         .         .         .63 

§  38.   Critical  Values 63-64 

§  39.    Fundamental  Theorem 64 

§  40.    Final  Tests 64-65 

§  41.    Illustrative  Examples  in  Extremes         ....  65-67 


Exercises  XIV,     Extremes 


Part  II        Rates 


§  42.    Time  Rates 
§  43.    Speed      . 


67-70 

70-90 

70 

71 


CONTENTS  XI 

PAGEk 

§  44.    Tangential  Acceleration 71 

§  45.    Second  Derivative.     Flexion 71-72 

Exercises  XV.     Second  Derivatives.     Acceleration     .         .  73-75 

§  46.   Concavity.     Points  of  Inflexion  ....  75 

§  47.    Second  Test  for  Extremes 75-76 

§  48.    Illustrative  Examples 76-77 

§  49.    Derived  Curves 77-79 

Exercises  XVI.     Flexion.     Derived  Curves         .         .         .  79-81 

§  50.    Angular  Speed 81 

§  51.    Angular  Acceleration 81-82 

§  52.    Momentum.     Force 82-83 

Exercises  XVII.     Time  Rates 83-85 

§  53.    Related  Rates 85-87 

Exercises  XVIII.     Related  Rates 88-90 

CHAPTER  V      REVERSAL  OF  RATES      INTEGRATION 

SUMMATION 91-129 

Part  I        Integrals  by  Reversal  of  Rates    .  91-109 

§  64.    Reversal  of  Rates        .         .  ....  91-92 

§  65.    Principle  Involved  m  §  54  .         .         .  .         .92 

§  56.    Illustrative  Examples 92-94 

Exercises  XIX.     Reversal  of  Rates 94-95 

§  67.    Integral  Notation 96-97 

Exercises  XX.     Notation.     Indefinite  Integrals  .         .         .  97-98 

§  58.    Fundamental  Theorem 99 

§  59.   Definite  Integrals 100-101 

Exercises  XXI.     Definite  Integrals 102-103 

§  60.    Area  under  a  Curve 103-104 

Exercises  XXII.     Area 105 

§  61.    Lengths  of  Curves 106-107 

§  62.    Motion  on  a  Curve.     Parameter  Forms      .         .        .  107 

§  63.    Illustrative  Examples 108-109 

Exercises  XXIII.     Length.     Total  Speed    ....  109 

Part  II         Integrals  as  Limits  of  Sums        .  110-120 

§  64.    Step-by-Step  Process 110-111 

§  65.    Approximate  Summation 111-112 

Exercises  XXIV.    Step-by-Step  Summation.     Approximate 

Results 112-113 


xu 


CONTENTS 


§  66.    Exact  Results.     Summation  Formula 
§  67.    Integrals  as  Limits  of  Sums 
§  68.     Water  Pressure  .... 

Exercises  XXV.     Integrals  as  Limits  of  Sums 

§69.    Volumes 

§  70.    Volume  of  Any  Frustum     . 

Exercises  XXVI.     Volumes  of  Solids.     Frusta 

§  71.   Cavalieri's  Theorem.     The  Prismoid  Formula 

Exercises  XXVII.     General  Exercises 


PAGES 

114-116 
116-117 
117-119 
119-120 
120-121 
121-123 
124-125 
125-127 
128-129 


CHAPTER   VI        TRANSCENDENTAL   FUNCTIONS 


130-173 


Pakt  I        Logarithms        Exponential  Functions    130-149 


§  72.   Necessity  of  Operations  on  Transcendental  Functions 

§  73.   Properties  of  Logarithms 

§  74.     Graphical  Representation 

Exercises  XXVIII.     Logarithms  and  Exponentials 
§  75.    Slope  of  y  =  logio  x  at  x  =  0.     [Modulus  M] 

§  76.    Differentiation  of  logio  x 

§77.    Differentiation  of  loga  X.     [Napierian  Base] 

§  78.    Illustrative  Examples 

Exercises  XXIX.     Logarithms 

§  79.    Differentiation  of  Exponentials 

§  80.    Illustrative  Examples 

Exercises  XXX.     Exponentials 

§81.    Compound  Interest  Law 

Example  1.    Work  in  Expanding  Gas         .     (142) 
Example  2.    Cooling  in  a  Moving  Fluid      .     (142-143) 
Example  3.     Bacterial  Growth  .         .         .     (143) 
Example  4.     Atmospheric  Pressure    .         .      (143) 
§82.   Percentage  Rate  of  Increase.     [Relative  Rates] 
Exercises  XXXI.     Compound  Interest  Law 
§  83.   Logarithmic  Differentiation.     Relative  Increase 
§  84.   Logarithmic  Methods  .... 

Exercises  XXXII.     Logarithmic  Differentiation 

Part  II        Trigonometric  Functions 

§  85.    Introduction  of  Trigonometric  Functions   . 
S  86.   Differentiation  of  Sines  and  Cosines  . 


130 

130-131 

131-132 

132-133 

133-134 

134 

135-136 

136-137 

137-138 

138-139 

139-140 

140-141 

141-143 


144 

144-146 

146-147 

147-148 

149 

150-173 

150 
150-153 


CONTENTS 


xiu 


PAGES 

§  87.    Illustrative  Examples 152-153 

Exercises  XXXIII.     Trigonometric  Functions  .        .        .  153-155 

§  88.    Simple  Harmonic  Motion 155-156 

§  89.   Relative  Acceleration 156 

§  90.   Vibration 157 

§91.    Waves 157-158 

Exercises  XXXIV.  Simple  Harmonic  Motion.   Vibrations  158-100 

§  92.    Damped  Vibrations 160-162 

Exercises  XXXV.     Damped  Vibrations     ....  162-163 

§  93.    Inverse  Trigonometric  Functions       ....  163-164 

§  94.    Integrals  of  Irrational  Functions       ....  164 

§  95.   Illustrative  Examples 164-165 

Exercises  XXXVI.     Inverse  Trigonometric  Functions      .  165-166 

§  96.   Polar  Coordinates 166-168 

Exercises  XXXVII.     Polar  Coordinates     ....  168-169 

§97.   Curvature 169-171 

Exercises  XXXVIII.     Curvature 171-172 

§  98.   Collection  of  Formulas 173 

CHAPTER   VII        TECHNIQUE         TABLES         SUCCES- 
SIVE  INTEGRATION 174-226 

Part  I        Technique  of  Integration  .         .  174-200 

§  99.    Question  of  Technique.     Collection  of  Formulas      .  174-175 
§  100.     Polynomials.     Other  Simple  Forms          .         .         .176 
§  101.    Substitution.     [Algebraic  and  Trigonometric]  .         .  176-177 
§  102.    Substitution  in  Definite  Integrals       .         .         .         .177 
Exercises  XXXIX.    Elementary  Integration.    Substitution  178-181 
§  103.    Integration   by   Parts.     [Algebraic   and  Transcen- 
dental]              181 

Exercises  XL.     Integration  by  Parts 182-183 

§  104.    Rational  Functions.     [Partial  Fractions]         .         .  184-186 

Exercises  XLI.     Rational  Functions 186-188 

§  105.    Rationalization  of  Linear  Radicals   ....  188-189 

§  106.    Quadratic  Irrationals 189-190 

Exercises  XLII.     Integrals  involving  Radicals.     [Trigono- 
metric Substitutions] 191-194 

§  107.   Elliptic  and  Other  Integrals 195 

§  108.    Binomial  Differentials 195-196 


XIV 


CONTENTS 


§  109.   General  Remarks.     [Tables.]   . 
Exercises  XLIII.     General  Integration. 


[All  Methods.] 


PAGES 

196 
197-200 


Part  II 


Improper  and  Multiple  Integrals       201-226 


§  110.    Limits  Infinite.     Horizontal  Asymptote  . 
§  111.    Integrand  Infinite.     Vertical  Asymptotes 

§  112.   Precautions 

Exercises  XLIV.     Improper  Integrals 

§  113.    Repeated  Integration 

§  114.    Successive  Integration  in  Two  Letters 

Exercises  XLV.     Successive  Integration 

§  115.    Double  Integrals        .... 

§  116.    Illustrative  Examples 

[A]  Volumes  by  Double  Integration     . 

[B]  Area  in  Polar  Coordinates     . 

[C]  Moment  of  Inertia  of  a  Thin  Plate 

[D]  Moment  of  Inertia  in  Polar  Coordinates 
Exercises  XLVI.     Double  Integrals     . 
§  117.    Triple  and  Multiple  Integrals    . 
Exercises  XLVII.     Multiple  Integrals 
§118.    Other  Applications.     Averages.     Centers  of  Gravity 
Exercises  XLVIII.     General  Problems  in  Integration 


(212) 
(212-213) 
(21.3-214) 
(214) 


201 

202-203 

203 

204-205 

206 

206-207 

208-209 

210-211 

211-214 


214-217 
217-218 
218-219 
219-220 
221-226 


CHAPTER   VIII        METHODS   OF   APPROXIMATION     .  227-280 

Part  I        Empirical  Curves        Increments     .  227-250 

§  119.    Empirical  Curves 227 

§  120.    Polynomial  Approximations 227 

§  121.    Review  of  Elementary  Methods         ....  227-229 

§  122.   Logarithmic  Plotting         .         .         .         .     -   .         .  229-230 

§  123.    Semi-Logarithmic  Plotting 230 

Exercises  XLIX.     Empirical  Curves.     Elementary  Methods  230-233 

§  124.    Method  of  Increments 233-236 

Exercises  L.     Empirical  Curves  by  Increments  .         .         .  236-238 

§  125.    Approximate  Integration 239-240 

§  126.   Integration  from  Empirical  Fornmlas        .         .         .  240-241 

§  127.   Derived  and  Integral  Curves 241-242 

Exercises  LI.     Approximate  Evaluation  of  Integrals  .  242-243 

5128.    Integrating  Devices.     [Planimeter.     Integraph]       .  243-246 


CONTENTS  XV 

PAGES 

§  129.    Tabulated  Integral  Values         .....  246-247 

Exercises  LII.     Integrating  Devices.     Numerical  Tables    .  248-250 

Part  II        Polynomial  Approximations         Series 

Taylor's  Theorem           .         .         .  250-280 

§  130.    Rolle's  Theorem 250 

§  131.    The  Law  of  the  Mean.     [Finite  Differences]     .         .  251 

§  132.    Increments.     [Small  Errors] 252-253 

Exercises  LIII.     Increments.     Law  of  the  Mean        .         .  253-254 

§  133.    Limit  of  Error 251-257 

§  134.    Extended  Law  of  the  Mean.     Taylor's  Theorem      .  257-259 

Exercises  LIV.     Extended  Law  of  the  Mean       .         .         .  259-260 

§  135.    Application  of  Taylor's  Theorem  to  Extremes          .  260-261 

Exercises  LV.     Extremes 261-263 

§  136.    Indeterminate  Forms.     [Form  0  h-  0]       .         .         .  263-265 

§137     Infinitesimals  of  Higher  Order           ....  265-266 

Exercises  LVI.     Indeterminate  Forms.     Infinitesimals       .  266-267 

§  138.    Double  Law  of  the  Mean 267-268 

§  139.    The  Indeterminate  Form  (»  h-  co.     Vertical 

Asymptotes 268-269 

§  140.    Other  Indeterminate  Forms 269-270 

Exercises  LVII.    Secondary  Indeterminate  Forms       .         .  270-271 

§  141.    Infinite  Series 271-273 

§  142.    Taylor  Series.     General  Convergence  Test       .         ,  273-275 

Exercises  LVIII.     Taylor  Series 275-276 

§  143.    Precautions  about  Infinite  Series       ....  276-279 

Exercises  LIX.     Infinite  Series 279-280 

CHAPTER   IX         SEVERAL   VARIABLES        PARTIAL 

DERIVATIVES        APPLICATIONS         GEOMETRY  281-344 

Part  I        Partial  Differentiation        Elementary 

Applications        ....  281-297 

§  144.    Partial  Derivatives             281-282 

§  145.    Technique 282 

§  146.    Higher  Paitial  Derivatives 282-283 

Exercises  LX.     Technique  of  Partial  Differentiation  .         .  283-284 

§  147.    Geometric  Interpretation 284-285 

§  148.    Total  Derivative 285-287 


CONTENTS 


§  149.    Elementary  Use 

§  150.    Small  Errors.     Partial  Differentials 
Exercises  LXI.     Total  Derivatives  and  Differentials  . 
§  151.    Significance  of  Partial  and  Total  Derivatives    . 

Example  1.  Isothermal  Expansion       .        .   (292-293) 

Example  2.  Adiabatic  Expansion         .        .   (293-294) 

Example  3.  Implicit  Equations.    Contour  Lines  (294) 

Example  4.  Flow  of  Heat  in  a  Metal  Plate     (294-295) 

Example  5.  Flow  of  Water  in  Pipes    .        .     (295-296) 

Exercises  LXII.     Applications  of  Total  Derivatives    . 

Part  II        Applicatbons  to  Plane  Geometry   . 


Evolute 
Evolutes 


§  152.    Envelopes 

§  153.    Envelope  of  >Jormals. 

Exercises  LXIII.     Envelopes. 

§  154.    Properties  of  Evolutes       .... 

§  155.    Center  of  Curvature  .... 

§  156.    Rate  of  Change  of  ^ 

§  157.   Illustrative  Examples        .... 

Exercises  LXIV.     Properties  of  Evolutes    . 

§  158.    Singular  Points 

§  159.   Illustrative  Examples        .... 

§  160.    Asymptotes 

§  161.    Curve  Tracing 

Exercises  LXV.     Singular  Points.     Asymptotes. 
Tracing 


Curve 


PAG! 

287-2 ■ 
288-2" 
290-2 
292-2  - 


296-297 

298-315 

298-300 

300-303 

303 

304-305 

305 

305-306 

306-307 

308-309 

309-310 

310-311 

311-313 

313 

313-314 


Part  III        Geometry  of  Space        Extremes  .     315-344 


162.    R^sum^  of  Formulas 


(a)  Distance  between  Two  Points 

(&)  Distance  from  Origin  to  {x,  y,  z) 

(c)  Direction  Cosines 

(d)  Angle  between  Two  Directions 

(e)  The  Plane      .... 
(/)  The  Straight  Line 

(f/)  Quadric  Surfaces  . 


§  163.    Loci  of  One  or  More  Equations 
Exercises  LXVI.     R6sum6  of  Solid  Geometry 


(315) 

(315) 

(315) 

(315) 

(316-317) 

(317 

(318) 


315-318 


318-319 
319-320 


CONTENTS 


XVU 


PAOBS 

§  164.    Tangent  Plane  to  a  Surface 321-322 

§  165.   Extremes  on  a  Surface.     [Least  Squares]         .        .  322-325 

§166.   Final  Tests 325-326 

Exercises  LXVII.     Tangent  Planes.    Extremes  .         .  327-329 

§  167.   Tangent  Planes.     Implicit  Forms      ....  329-330 

§  168.   Line  Normal  to  a  Surface 330 

§  169.    Parametric  Forms  of  Equations         ....  330-332 

§  170.    Tangent  Planes  and  Normals.     Parameter  Forms    .  332-333 

Exercises  LXVIII.     Equations  not  in  Explicit  Form  .         .  333-334 

§  171.    Area  of  a  Curved  Surface 334-335 

Exercises  LXIX.     Area  of  a  Surface 336 

§  172.    Tangent  to  a  Space  Curve 337 

§  173.   Length  of  a  Space  Curve 338 

Exercises  LXX.   Tangents  to  Curves.     Lengths  .         .  338 

Exercises  LXXI.     General  Review.     Several  Variables       .  339-344 


CHAPTER   X        DIFFERENTIAL   EQUATIONS 


345-383 


Part  I 


Ordinary  Differential  Equations  of 
THE  First  Order      .         .         .         . 


§  174.   Reversal  of  Rates 345 

§  175.   Other  Reversed  Problems 345-346 

§  176.    Determination  of  the  Arbitrary  Constants        .        .  346-347 
§  177.    Vital  Character  of  Inverse  Problems         .         .         .  347-348 
§178.    Elementary  Definitions.   Ordinary  Differential  Equa- 
tions         348 

§  179.   Elimination  of  Constants 348-350 

§  180.    Integral  Curves 350-351 

Exercises  LXXII.     Elimination.     Integral  Curves      .         .  351-352 

§  181.   General  Statement 352-353 

§  182.    Type  I.     Separation  of  Variables     .        .        .        .353 

§  183.   Type  II.     Homogeneous  Equations  ....  354-355 

Exercises  LXXIII.     Separation  of  Variables       .        .         .  355-^356 

§  184.   Tjije  III.     Linear  Equations 356-358 

§  185.    Extended  Linear  Equations 358 

Exercises  LXXIV.     Linear  Equations  ....  359 

§  186.     Other  Methods.     Non-linear  Equations   .         .         .  359-360 

Exercises  LXXV.     Miscellaneous  Exercises        .        .        .  360-362 


XVlll 


CONTENTS 


Part  II 


Ordinary  Differential  Equations  of 
THE  Second  Order  .... 


Order 


Part  III         Generalizations 

§  192.    Ordinary  Equations  of  Higher  Order 

§  193.   Linear  Homogeneous  Type 

§  194.   Non-homogeneous  Type     . 

Exercises  LXXX.     Linear  Equations  of  Higher 

§  195.     Systems  of  Differential  Equations 

§  196.   Linear  Systems  of  the  First  Order 

§197.     dx/P=dy/Q  =  dz/R 

Exercises  LXXXI.     Systems  of  Equations 

§  198.    Partial  Differential  Equations 

§  199.    Relation  to  Systems  of  Ordinary  Equations 

Exercises  LXXXII.     Partial  Differential  Equations 


363-374 


§  187.   Special  Types 

§  188.   Type  I :  d-s/df-  =  ±  k'^s 

Exercises  LXXVI.     Type  I 

§189.   Type  II.      Homogeneous  Linear  —  Constant  Coeffi- 
cients      

Exercises  L  XXVII.     Type  II.     Linear  Homogeneous 
§  190.   Type  III.     Non-homogeneous  Equations  . 
Exercises  LXXVIII.     Non-homogeneous.     Type  III . 
§191.   Type  IV.     One  of  the  quantities  a;,  j/,?/' absent 

(a)  Type  IV  (a) :  <f>  (y")  =  0       .          .          .     (371) 
(6)  Type  IV  (6):  0  (.7;,  2/',?/")  =  0         .          .     (371-373) 
(c)    Type  IV  (c) :  4>  (V,  V' ,  V")  =0         .          •     (373-374) 
Exercises  LXXIX.     Type  IV 374 


363-365 


368-369 
369-370 
371 
371-374 


375-383 

375 

375-377 

377-378 

378-379 

379 

379 

379-381 

381-382 


TABLES 

[Note  page  numbers  in  italic  numerals] 

TABLE   I        SIGNS   AND   ABBREVIATIONS        .  1-3 

TABLE   II        STANDARD   FORMULAS.         .         .  3-16 

A.  Exponents  and  Logarithms 3 

B.  Factors 4 

C.  Solution  of  Equations.     Determinants 4-5 


CONTENTS  xix 

PAGES 

D.  Applications  of  Algebra 5-6 

E.  Series.     [Special.    Theorems  of  Taylor  and  Fourier]      .         .  7-8 

F.  Geometric  Magnitudes.     Mensuration S-11 

G.  Trigonometric  Relations 12-13 

H.    Hyperbolic  Functions 13-14 

I.   Analytic  Geometry _.  14-15 

J.    Differential  Formulas * .  15-16 

Table  III        Standard  Curves  .         .         .  17-32 

A.  Curves  y  =  .r».     [Chart  of  Entire  Family]      ....  17- IS 

B.  Logarithmic  Paper.     Curves  y  =  x",  y  ~  k.r"  .        .         .         .IS 

C.  Trigonometric  Functions 19 

D.  Logarithms  and  Exponentials.     [Bases  10  and  r]            .         .  19 

E.  Exponential  and  Hyperbolic  Functions 20 

F.  Harmonic  Curves.     [Simple  and  Compound]          .         .         .  31-22 

G.  The  Roulettes.     [Cycloid,  Trochoids,  etc.]     ....  22-24 
H.   The  Tractrix 24 

I.    Cubic  and  Quartic  Curves.     Contour  Lines    ....  25-27 

J.    Error  or  Probability  Curves 28 

K.    Polynomial  Approximations.     [Taylor  and  Lagrange]            .  28-29 

L.   Trigonometric  Approximations.     [Fourier]    .         .         .         .  29 

M.    Spirals 30-31 

N.   Quadrlc  Surfaces 31-32 

Table  IV        Standard  Integrals          .         .  33-48 

A.  Fundamental  General  Formulas 33 

B.  Integrand  Rational  Algebraic 34-36 

C.  Integrand  Irrational 37-39 

(a)  Linear  radicals  r  =  Vox+T        .         .         .        {37) 

(b)  Quadratic  radicals (3T-39) 

D.  Binomial  Differentials — Reduction  Formulas         .        .         .39 

E.  Integrand  Transcendental 39-44 

(a)   Trigonometric (,39-42) 

(6)   Trigonometric  —  Algebraic  .  (42-43) 

(c)  luverse  Trigonometric         ....        {43) 
{(i)   Exponential  and  Logarithmic      .         .         .        (43-44) 

F.  Some  Important  Definite  Integrals 44-4^ 

G.  Approximation  Formulas 45-46 

H.    Standard  Applications  of  Integration 4(>-48 


XX  CONTENTS 

PA6KB 

Table  V        Numerical  Tables  .         .        .     49-58 

A.  Trigonometric  Functions.     [Values  and  Logarithms]     .         .  49 

B.  Common  Logarithms 50-51 

C.  Exponential  and  Hyperbolic  Functions.     Natural  Logarithms  52 

D.  Elliptic  Integral  of  the  First  Kind 53 

E.  Elliptic  Integral  of  the  Second  Kind 53 

F.  Values  of  n(p)  =  r(p  +  1).     Gamma  Function     .         .         .54 

G.  Values  of  the  Probability  Integral 54 

H.    Values  of  the  Integral  |       (e*/a;)dx 54 

I.  Eeciprocals  ;  Squares ;  Cubes 55 

J.  Square  Roots ,         .        .     56 

K.  Radians  to  Degrees 56 

L.  Important  Constants 57 

M.  Degrees  to  Radians 57 

N.  Short  Conversion  Table  and  Other  Data  .        ,        .        .68 

IXDEX 59 


Be  not  the  first  by  whom  the  new  is  tried, 
Nor  yet  the  last  to  lay  the  old  aside." 

—  POPK. 


THE   CALCULUS 


CHAPTER  I 

FUNCTIONS 

1.  Dependence.  There  are  countless  instances  in  -which  one 
quantity  depends  upon  another.  The  speed  of  a  body  falling 
from  rest  depends  upon  the  time  it  has  fallen.  One's  income 
from  a  given  investment  depends  upon  the  amount  invested 
and  the  rate  of  interest  realized.  The  crops  depend  upon  rain- 
fall, soil  fertility  and  proper  cultivation. 

In  mathematics  we  usually  deal  with  quantities  that  are 
definitely  and  completely  determined  by  certain  others.  Thus 
the  area  ^  of  a  square  is  determined  precisely  when  the  length 
s  of  its  side  is  given  :  A  =  s^;  the  volume  of  a  sphere  is  4  tti^/S  ; 
the  force  of  attraction  between  two  bodies  is  k  ■  m  •  m' /d^,  where 
m  and  m'  are  their  masses,  d  the  distance  between  them,  and  k 
a  certain  number  given  by  experiment.  The  Calculus  is  the 
stvidy  of  the  relations  between  such  interdependent  quantities, 
with  special  reference  to  their  rates  of  change. 

2.  Variables.  Constants.  Functions.  A  quantity  which 
may  change  is  called  a  variable.  The  quantities  mentioned  in 
§  1,  except  k  and  tt,  are  examples  of  variables. 

A  quantity  which  has  a  fixed  value  is  called  a  constant.  Ex- 
amples of  constants  are  ordinary  numbers :  1,  V2,  —  7,  2/3,  tt, 
30°,  log  5,  and  the  number  k  in  §  1. 

If  one  variable  y  depends  on  another  variable  x,  so  that  ?/  is 
determined  when  x  is  known,  y  is  said  to  be  a  function  of  x. 

B  1 


FUNCTIONS 


[I,  §2 


The  variable  x,  thus  thought  of  as  determining  the  other,  is 

called  the  independent  variable ;  the  other  variable  y  is  called 
the  dependent  variable.  Thus,  in  §  1,  the 
area  ^  of  a  square  is  a  function,  A  =  s^,  of 
the  side  s. 

In  Algebra  we  learn  how  to  express  such 
relations  by  means  of  equations. 

In  Analytic  Geometry  such  relations  are 
represented  graphically.  For  example,  if 
the  principal  at  simple  interest  is  a  fixed 
sum  p  and  if  the  interest  rate  r  also  is 
fixed,  then  the  amount  a,  of  principal  and 
interest,  varies  solely  with  (is  a  function  of) 

the  time  t  that  the  principal  has  been  at  interest.     In  fact,  Up 

=  100  and  r  =  6%, 


- 

- 

^ 

' 

: 

- 

?> 

- 

-K 

0 

ie 

- 

1 

1 

1 

. 

<  (Years) 
Fig.  1. 


a=p +  2:)tr  =  100  + 6  t. 


This  is  represented  graphically  in  Fig.  1. 
parts  of  a  day  are  neglected. 

The  relation  ^  =  s*  of  §  1  is  repre- 
sented in  Fig.  2. 


EXERCISES  I.  — FUNCTIONS  AND  GRAPHS 

Represent  graphically  the  following  :  — 

1.  a  =  100  +  3  «,  a  =  300  +  4  «,  a  =  150 
+  7«. 

2.  The  number  of  feet  /  in  terms  of  the 
number  of  yards  ?/  in  a  given  length  is  given 
by  the  equation  f  =  3y. 


In  practice  fractional 


/ 

A 

\ 

(8 

3.fe 

et)J 

1 

\ 

\ 

\ 

1 

4=s 

2 

k 

1 

L 

> 

s 

(fe 

et) 

0      12    3    4 
Fig.  2. 


3.  The  temperature  in  degrees  Fahrenheit,  F,  is  32  more  than  9/5  the 
temperature  in  degrees  Centigrade,  C. 

4.  The  distance  s  that  a  body  falls  from  rest  in  a  time  t  is  given  by 
s  =  16  t'^.     (Measure  t  horizontally  and  s  vertically  dovirnward.) 


I,  §  3]  GR.IPHS      NOTATION 


5. 

(a)  y  =  «2  +  3x+l. 

(4)  y  =  2x'-6x. 

(c)  y  =  oi?+2. 

w,=^;^. 

«^=l^- 

Wy.-±^ 

6.  The  volume  v  of  a  fixed  quantity  of  gas  at  a  constant  temperature 
varies  inverse!}'  as  tlie  pressure  p  upon  the  gas. 

7.  The  amount  of  6 1.00  at  compound  interest  at  10  %  per  annum  for 
t  years  is  a  =(1  +  1/10)*. 

8.  The  area  A  of  an  equilateral  triangle  is  a  function  of  its  side  s. 
Determine  this  function,  and  represent  the  relation  graphically.  Express 
the  side  in  terms  of  the  area. 

9.  Determine  the  area  a  of  a  circle  in  terms  of  its  radius  r.  Deter- 
mine the- radius  in  terms  of  the  area. 

10.  The  radius,  surface,  and  volume  of  a  sphere  are  functionally  re- 
lated. Find  the  equations  connecting  each  pair.  Also  express  each  of 
the  three  as  a  function  of  the  circumference  of  a  great  circle  of  the  sphere. 

11.  The  area  A  bounded  by  the  straight  line  y  =  ax  +  b,  the  ordinate 
y,  and  the  axes,  is  a  function  of  x.  Determine  it ;  and  also  express  y  as 
a  function  of  the  area. 

3.  The  Function  Notation.  A  very  useful  abbreviation  for 
functions  consists  in  writing /(^)  (read /of  x)  in  place  of  the 
given  expression. 

Thus  iff{x)  =  x^  +  3x  +  l,we  may  write  /(2)  =  2^  +  3  •  2  + 
1  =  11,  that  is,  the  value  of  u^  +  Sx  +  l  when  x  =  2  is  11. 
Likewise  /(3)  =  19,  /(-  1)  =  - 1,  /(O)  =  1,  and  so  on.  /(«)  = 
a^  +  3  a  + 1.    fill  +  v)  =  (u-\-  vf  +  3(w  +  v)  + 1. 

Other  letters  than  /  are  often  used,  to  avoid  confusion,  but 
/is  used  most  often,  because  it  is  the  initial  of  the  word  func- 
tion. Other  letters  than  x  are  often  used  for  the  variable.  In 
any  case,  given  f{x),  to  find  /(a),  simply  substitute  a  for  x  in 
the  given  expression. 

EXERCISES  II.  —  SUBSTITUTION       FUNCTION  NOTATION 

1.  If  /(x)  =  x2  -  5  X  +  2  find  /(I),  /(2),  /(3),  /(4),  /(O),  /(-  1), 
/(-2).    From  these  values  (and  others,  if  needed)  draw  the  graph  of 


4  FUNCTIONS  [I,  §  3 

the  curve  y  =/(.r).     Mark  its  lowest  point,  and  estimate  the  values  of  jt 
and  y  there. 

2.  Proceed  as  in  Ex.  1  for  each  of  the  following  functions,  using  the 
function  notation  in  calculating  values ;  mark  the  highest  and  lowest 
points  if  any  exist,  and  estimate  the  values  of  x  and  y  at  these  points. 

(a)x3-2x  +  4.     (6)3x2_2x  +  l.     (c)^^.     (d)-^+-^. 

2x  —  3  x  +  1      X  —  1 

(e)  y  =  sin  x,  taking  x  =ir/Q,  tt/^,  t/2,  3  7r/4,  tt,  0,  -  7r/2. 

(/)  j/  =  logiox,  taking  x  =  1,  2,  10,  1/10,  1/100. 

3.  If  fix)  =  X*  -  6  x3  +  3  x2  -  2  X  +  3,  calculate  /(I),  /(4),  /(5). 
Hence  show  that  one  solution  of  the  equation  /(x)  =  0  is  x  =  1 ;  and 
that  another  solution  lies  between  4  and  5. 

(This  work  is  simplified  by  using  the  theorem  that  /(a)  is  equal  to  the 
remainder  obtained  by  dividing  f{x)  by  [x—a);  and  by  using  synthetic 
division.) 

4.  If /(x)  =  2x2-3x  +  5,  show  that /(a)  =  2a2_3rt4-5,/(m  +  M) 
=  2  (m  +  7i)2  _  3  (m  +  n)  +  5  ;  find  f{a  -  b),  f{a  +  2b),  fia/b). 

5.  If  /(x)  =  x2  4-  3  and  0  (x)  =  3  x  +  1,  show  that  f(l)  =  <P  (1)  and 
/(2)  =  0(2).     Show  that/(3)  >0(3).     Draw  y  =/(x)  and  y  =  0(x). 

6.  In  Ex.  5,  draw  the  curve  y  =f{x)  —  <p  (x).  Mark  the  points 
where  /(x)  —  ^  (x)  =  0.     Mark  the  lowest  point. 

7.  If  /(x)  =  -  2  x2  +  1  and  <p  (x)  =  x^ -\-2  x  +  4,  find  the  value  for 
which  /(x)  =  0  (x)  by  use  of  f(x)  —  <p  (x).  Sketch  all  of  the  curves 
j/=/(x),  y  =  0(x),  J/=/(x)-0(x). 

8.  If/(x)  =  sinx  and  0  (x)  =  cos  x,  show  that  [/(x)]2  +  [0  (x)]2 
=  1  ;  fix)  --  0  (x)  =  tan  x  ;  /(x  +  y)  =fix)  0  (?/)  +  fiy)  0  (x) ;  0  (x  +  ?/) 
=  ?;/(x)=:0(7r/2-x);0(x)=/(7r/2-x)  =  -0(^-x);/(-x)  =  -/(+x): 
0(-x)  =  0(x). 

9.  If  fix)  -  logio  X,  show  that 

n^)+fiy)=f(x-y);  /(x2)=2/(x); 

/(m/n)  -fin/m)  =  2/(m)  -  2/(n);        /(m/n)  +/(n/»i)  =  0. 

10.  If /(x)  =  tan  X,  0  (x)  =  cos  x,  draw  the  curves  y=fix),  y  =<t>  (x), 
2/  =/(x)  —  0  (x).  Mark  the  points  where  /(x)  =  0  (x)  and  estimate  the 
values  of  x  and  y  there. 

11.  Taking  /(x)  =  x^,  compare  the  graph  of  y=fix)  with  that  of 
y  -fix)  +  1,  and  with  that  of  y  =fix  +  1). 


I,  §3]  GRAPHS      NOTATION  5 

12.  Taking  any  two  curves  y=f(x),  y  =  <p{x),  how  can  you  most 
easily  draw  y  =f{x)  -<p{x)?    y  =f{x)  +  <t>  (.<)  ?     Draw  y^x^  +l/x. 

13.  How  can  you  most  easily  draw  y  =f{x)  +  5?  y  =f{x  +  5)  ?  as- 
suming that  y  =/(x-)  is  drawn. 

14.  Draw  y  =  x-  and  show  how  to  deduce  from  it  the  graph  of 
y  =  2x- ;  the  graph  oi  y  =  —  a;-. 

Assuming  that  y  =/(x)  is  drawn,  show  how  to  draw  the  graph  of 
2/  =  2/(.r);  that  of  y=-f{x). 

15.  From  the  graph  of  y  =  x"^,  show  how  to  draw  the  graph  of 
y  =  (2  a;)-;  that  of  y  =  x^  +  2  ;  that  of  y  =(a;+2)2  ;  that  of  y={2x-SyK 

16.  What  change  is  made  in  a  curve  if  x,  in  the  equation,  is  replaced 
by  -  .r  ?  if  y  by  -  2/  ?  if  both  things  are  done  ?  Compare  the  graphs  of 
y=fix),  y=f{-x),  -y=f{x);  y=.2f(x);  y=/(x)  +  2. 

17.  What  change  is  made  in  a  curve  if  x  is  replaced  by  2  x,  3  x,  x/2  ? 
Compare  the  graphs  of  y=f{x),  y=f{2x),  t/=/(3x),  y=f{x/2); 
y=f{x  +  2). 

18.  What  is  the  effect  upon  a  curve  if,  in  the  equation,  x  and  y  are 
interchanged  ?     Compare  the  graphs  of  y  =/(x),  x  =/(?/). 

19.  Plot  the  following  curves  :  (a)  y+2  =  sin  (3x  +  2),  (6)  y=x+s\nx, 
(c)  j/  =  2*  — sinx,  (d)  2/=2='cosx,  (e)  3x  +  4  y  =  4  sin  (4  x  — 3?/), 
(/)  ?/=  (cosx)/(2x  + 3),    (gr)  sin  2/ =  cos  2  X,    W  y  =  logo  (x^  +  1). 

20.  In  polar  coordinates  (r,  0),  what  change  is  made  in  a  curve  if,  in 
the  equation,  d  is  replaced  by  2  ^,  if  r  is  replaced  by  2  r  ? 

21.  What  change  in  6  is  equivalent  to  a  change  in  the  sense  of  r. 

22.  From  the  graph  of  r=f{d)  derive  those  of  («)  r=f(2d), 
{b)r  =  2f{9),  (c)  r=f{-e),  (d)  r=-f{e),  (e)  r  +  l=f{d), 
Lnr=fie+l),   (r/)r+l=/(5  +  2). 

Take,  for  example,  /((?)=  1,  f{d)  =  e,  /(^)  =  sin(?,  f{e)  =  2d,  f{e) 
=  arc  tan  d,  and  draw  the  variations  from  the  original  gi-apli. 

23.  Plot  the  following :  (a)  r  =  2  +  3  cos  ^,  (6)  r  =  3  +  2  cos  ^,  (c) 
r  =  2  +  2costf,  (d)  r  =  29,  (e)  r'^  =  ad,  (/)  0  =  2%  (g)  e^  =  ar, 
{h)e  =  smr,  {i)e  =  cosr,  (j)^  =  tanr,  (A)  r  =  sec  (tf-o),  (I)  e=secr. 

24.  Show  how  to  obtain  the  graph  of  y  =  ^  sin  (at  +  b)  by  suitable 
modification  of  the  simple  sine-curve  y  =  sin  t. 

25.  Draw  the  graphs  from  the  following  equations  :  (a)  2  s  =  e«  +  e-', 
(b)  2s  =  e*-e-«,  (c)  s  =  {e*  +  e-')/(et  -  e-'),  (d)  s  =  sin  (  +  sin  2  «, 
(e)  s  =  smt  +  e-«  sin  2  t.  Take  e  =  2.7,  and  use  logarithms  in  com- 
putations. 


CHAPTER  II 


RATES    LIMITS    DERIVATIVES 


4.  Rate  of  Increase.  Slope.  In  the  study  of  any  quantity, 
its  rate  of  increase  (or  decrease),  when  some  related  quantity 
changes,  is  very  important  for  auy  complete  understanding. 
Thus,  the  rate  of  increase  of  the  speed  of  a  boat  when  the 
power  applied  is  increased  is  a  fundamental  consideration. 
Graphically,  the  rate  of  increase  of  y  with  respect  to  x  is 
shown  by  the  rate  of  increase  of  the  height 
of  a  curve.  If  the  curve  is  very  flat,  there 
is  a  small  rate  of  increase ;  if  steep,  a  large 
rate. 

The  steepness,  or  slope,  of  a  curve  shows 
the  rate  at  which  the  dependent  variable  is 
increasing  with  respect  to  the  independent 
variable. 

When  we  speak  of  the  slope  of  a  curve 
at  any  point  P  we  mean  the  slope  of  its  tan- 
gent at  that  point.  To  find  this,  we  must 
start,  as  in  Analytic  Geometry,  with  a  secant 
through  P. 

Let  the  equation  of  the  curve.  Fig.  3,  be 
y  =  X-,  and  let  the  point  P  at  which  the  slope  is  to  be  found,  be 
the  point  (2,  4). 

Let  Q  be  any  other  point  on  the  curve,  and  let  Ax  represent 
the  difference  of  the  values  of  x  at  the  two  points  P  and  Q* 

*  Ax  may  be  regarded  as  an  abbreviation  of  the  phrase,  "  difference  of  the 
a;'s."  The  quotient  of  two  such  differences  is  called  a  difference  quotient. 
Notice  particularly  that  Ax  does  not  mean  A  X  x.  Instead  of  "  difference  of 
the  x's,"  the  phrases  "change  in  x  "  and  "  increment  of  x  "  are  often  used. 


1 

T 

iT 

clUt 

n 

i 

y=x^l'^ 

IS 

i        A     '• 

\        W     ; 

^  yikt^,B 

0V$  3  i 

II,  §  4]  RATES  7 

Then,  in  the  figure,    0/1  =  2,   AB  =  ^x,   and  OJ5  =  2  +  Aar. 

Moreover,  since  y  =  x-  at  every  point,  the  value  of  ?/  at  Q  is 
BQ=(2  +  Axy. 

The  slope  S  of  the  secant  PQ  is  the  quotient  of  the  differ- 
ences A?/ and  Aa; :  Us-     \  ^M^^-^- 
^  =  tanZ3/PQ  =  ^  =  M=(2  +  Aa;)--4^ 
Ax      PJi                Aa; 

The  slope  ?ii  of  the  tangent  at  P,  that  is  tan  Z  MPT,  is  the 
limit  of  the  slope  of  the  secant  as  Q  approaches  P. 

The  slope  of  the  secant  is  the  average  slope  of  the  curve  between  the  points 
P  and  Q.  The  slope  of  the  curve  at  the  single  point  P  is  the  limit  of  this 
average  slope  as  Q  approaches  P. 

But,  since  *S  =  4  +  A.r,  it  is  clear  that  the  limit  of  aS  as  Q  ap- 
proaches P  is  4,  since  A.c  approaches  zero  when  Q  approaches 
P;  hence  the  slope  m  of  the  curve  is  4  at  the  point  P. 

At  any  other  point  the  argument  would  be  similar.  If  the 
coordinates  of  P  are  (a,  a-),  those  of  Q  would  be  [(a  +  Aa;), 
(a  +  Aa;)2] ;  and  the  slope  of  the  secant  would  be  the  difference 
quotient  A?/  -=-  Aa; : 

^^Ay^(a  +  Axf-a^^2aAa;  +  A^^2a  +  Aa;. 
Aa;  Aa;  Aa; 

Hence  the  slope  of  the  curve  at  the  point  (a,  a-)  is  * 

m  =  lim  S  =  lim  Ay /Ax  =  lim  (2  a  +  Aa;)  =  2  a. 

Ax=0  Ai=0  Ax=0 

On  the  curve  y  =  a?,  the  slope  at  any  point  is  numerically  twice 
the  value  of  x. 

When  the  slope  can  be  found,  as  above,  the  equation  of  the 
tangent  at  P  can  be  written  down  at  once,  by  Analytic  Geome- 
try, since  the  slope  m  and  a  point  (a,  h)  on  a  line  determine 
its  equation : 

(y  —  6)=  m(a;  —  a). 

Hence,  in  the  preceding  example,  at  the  point  (2,  4),  where 
we  found  m  =  4,  the  equation  of  the  tangent  is 

*  Read  "  Ax  =  0  "  "  as  Ax  approaches  zero."  A  detailed  discussion  of  limita 
is  given  in  §  10,  p.  16. 


8  DERIVATIVES  [II,  §  5 

(2/  —  4)  =  4  (a;  —  2),     or     4  a;  —  ?/  =  4. 
At  the  point  (a,  oF)  on  the  curve  y  =  x^,  we  found  m  =  2  a ; 
hence  the  equation  of  the  tangent  there  is 

(?/  —  a?)  =  2  a{x  —a),     or     2ax  —  y  =  a^. 

5.  General  Rules.  A  part  of  the  preceding  work  holds  true 
for  any  curve,  and  all  of  the  work  is  at  least  similar.  Thus, 
for  any  curve,  the  slope  is 

m  =  lim  S  =  lim(Ai//A.i;) ; 

Ai=0  Ai=0 

that  is,  the  slope  m  of  the  curve  is  the  limit  of  the  differenee  quo- 
tient Ay/Ax. 

The  changes  in  various  examples  arise  in  the  calculation  of 
the  difference  quotient,  Ay  -7-  Ax,  or  S. 

This  difference  quotient  is  alvmys  obtained,  as  above,  by  find- 
ing the  value  ofyat  Q  from  the  value  of  x  at  Q,  from  the  equa- 
tion of  the  curve,  then  finding  Ay  by  subtracting  from  this  the 
value  of  y  at  P,  and  finally  forming  the  difference  quotient  by 
dividing  Ay  by  Ax. 

6.  Slope  Negative  or  Zero.  If  the  slope  of  the  curve  is 
negative,  the  rate  of  increase  in  its  height  is  negative,  that  is, 
the  height  is  really  decreasing  with  respect  to  the  independent 
variable.* 

If  the  slope  is  zero,  the  tangent  to  the  curve  is  horizontal. 
This  is  what  happens  ordinarily  at  a  highest  point  (maximum) 
or  at  a  lowest  point  (minimum)  on  a  curve.t 

Example  1 .  Thus  the  curve  y  =  x^,  as  we  have  just  seen,  has,  at  any 
point  a:  =  a,  a  slope  m  =  2  a.    Since  m  is  positive  when  a  is  positive,  the 

*  Increase  or  decrease  in  the  height  is  always  measured  as  we  go  toward 
the  right,  i.e.  as  the  independent  variable  increases. 

t  A  maxinmm  need  not  be  the  highest  point  on  the  entire  curve,  but  merely 
the  highest  point  in  a  small  arc  of  the  curve  about  that  point.  See  §  37.  p.  63. 
Horizontal  tangents  sometimes  occur  without  any  maximum  or  any  minimum. 
See  §  38,  p.  63. 


11,  §  6] 


SLOPES  OF   CURVES 


9 


curve  is  vising  on  the  right  of  the  origin  ;  since  m  is  negative  when  a  is 
negative,  the  curve  is  f;illing  (tliat  is,  its  height  y  decreases  as  x  increases) 
on  the  left  of  the  origin.  At  the  origin  m  =  0  ;  the  origin  is  the  lowest 
point  (a  minimum)  on  the  curve,  because  the  curve  falls  as  we  come  toward 
the  origin  and  rises  afterwards. 

Example  2.     Find  the  slope  of  the  curve 
(1)  2/  =  x2  +  3a:-5 

at  the  point  where  x  =  —  2  ;  also  in  general  at  a  point  x  =  a.  Use  these 
values  to  find  the  equation 


of  the  tangent  at  a;  =  2  : 
tangent   at   any  point ; 


1/= 

X- 

*■  6 

r  - 

j 

0 

i 

\ 

\ 

/ 

1 

\ 

\ 

/ 

\ 

^ 

/l 

^U 

\ 

1 

1 

^ 

/ 

^ 

\ 

Az 

M 

\ 

T 

the 

the 
maximum  or  minimum 
points  if  any  exist. 

When  X  =  —  2,  we  find  y 
=  -  7,  (P  in  Fig.  4) ;  taking 
any  second  point  Q,  (—  2  + 
Ax,  —  7+Ay),  its  coordi- 
nates must  satisfy  the  given 
equation,  therefore 

(2)  -7  +  A2/  =  (-2  +  Aa;)2 
+  3(-2  + A.r)-5, 

or 

(3)  Ay  =  -  4  Ax  +Ax^  + 
3  Ax  =  —  Ax  +  Ax^, 

where  Ax^  means  the  square 
of  Ax.  Hence  the  slope  of 
the  secant  PQ  is  ^^^-  *• 

(4)  S  =  Ay /Ax  =  -  1  +  Ax. 

The  slope  m  of  the  curve  is  the  limit  of  S  as  Ax  approaches  zero  ;  i.e. 

(5)  m  =  lim  S=  lim  ^^=  lim  (_  1  +  Ax)  =  -  1. 

Ai=0  Ax=0  Ax        Ax  =  0 

It  follows  that  the  equation  of  the  tangent  at  (—  2,  —  7)  is 
(<>)  (2/  +  7)  =  - ](x  +  2),  or  x  +  2/  +  9  =  0. 

Likewise,  if  we  take  the  point  P  (a,  5)  in  any  position  on  the  curve 
whatsoever,  the  equation  (1)  gives 
(7)  ft  =  a2  +  3(j_5. 

Any  second  point  Q  has   coordinates    (a  +  Ax,  b  +  Ay)  where  Ax  and 


10 


DERIVATIVES 


[II,  §  6 


Ay  are  the  differences  in  x  and  in  ?/,  respectively,  between  Pand  §.  Since 
§  also  lies  on  the  curve,  these  coordinates  satisfy  (1)  : 

(8)  !>  +  A2/=(a  + Aa:)2  +  3(a  + Ax)-5. 
Subtracting  the  equation  (7)  from  (8), 

Ay  =  2  aAx  +  Ax"^  +  3  Ax,   vrhence    S  =  Ay /Ax  =  (2  a  +  3)  +  Ax, 
and 

(9)  m  =  lim  S=  lim  ^  =  lim  [(2 a  +  3)  +  Ax]  =  2 a  +  3. 

Ai=0  Aj^MjAX         Aj^  =  0 

Therefore  the  tangent  at  («,  6)  is 

(10)  2/-(a2  +  3a-5)  =  (2a  +  3)(x-a),  or  (2a  +  3)x-?/  =  a2  +  5. 
From  (9)  we  observe  that  to=0,  when  2  a  +  3  =  0,  i.e.  when  a  =  —  3/2. 

For  all  values  greater  than  —3/2,  m=(2a  +  3)  is  positive;  for  all 
values  less  than  —  3/2,  m  is  negative.  Hence  the  curve  has  a  minimum 
at  (—3/2,  —29/4)  in  Fig.  4,  since  the  curve  falls  as  we  come  toward 
this  point  and  rises  afterwards. 

Example  3.  Consider  the  curve  y  =  x^  —  12x  +  7.  If  the  value  of  x 
at  any  point  P  is  a,  the  value  of  ?/  is  a^  —  12  a  +  7.  If  the  value  of  x  at 
Qis  a  +  Ax,  the  value  of  ?/  at  (^  is  (a  +  Ax)^  _  12  (a  +  Ax)  +  7. 

Hence 

c,_Ay  _[(a  +  Ax)«  -  12  (g  +  Ax)  +  7]  -  [g^  _  12  g  +  7] 


Ax 


Ax 


IfBiPilH 

:|::::;:::::::..|^',   \':    i|ii    \\\y  {\\\ 

m^^\f4b 

^^dmmM 

^:::::a.LH^-  .in-  ;io,'^i\<fii  i  ni 

=  (3  g2  +  3  gAx  +  Ax")-  12, 
and 

3a2-12. 


lim  ^: 
Ai=o  Ax 


For  example,  if  x  =  1,  y 
=  —  4  ;  at  this  point  (1,  —  4) 
the  slope  is  3  •  l'^  -  12  =—  9 ; 
and  the  equation  of  the  tan- 
gent is 

(y  +  4)  =  -9(x-l),  or 

9x  +  ?/-5  =  0. 
Since  3g2  — 12  is  negative 
when  g-  <  4,  the  curve  is  fall- 
ing when  g  lies  between  —  2 
*'^'^-  ^-  and   +  2.     Since  3  cfi  -  12  is 

positive  when  a2>4,  the  curve  is  rising  when  x<—  2  and  when  x>+  2. 
At  X  =  ±  2,  the  slope  is  zero.     At  x  =  +  2  there  is  a  minimum  (see  Fig. 


II,  §6]  SLOPES  OF  CURVES  11 

5),  since  the  curve  is  falling  before  this  point  and  rising  afterwards.  At 
x=  —  2  there  is  a  maximum.  At  x=+  2,  y  =(2)8  —  12  ■  2  +  7  =—  0, 
which  is  the  lowest  value  of  y  near  that  point.  At  x  =  —  2,  y  =  23,  the 
highest  value  near  it. 

This  information  is  quite  useful  in  drawing  an  accurate  figure.  We 
know  also  that  the  curve  rises  faster  and  faster  to  the  right  of  x  =  2. 
Draw  an  accurate  figure  of  your  own  on  a  large  scale. 

EXERCISES   III.  — SLOPES   OF   CURVES 

1.  Find  the  slope  of  the  curve  y  =  x'^  +  2  at  the  point  where  x  =  1. 
Find  the  equation  of  the  tangent  at  that  point.  Verify  the  fact  that  the 
equation  obtained  is  a  straight  line,  that  it  has  the  correct  slope,  and  that 
it  passes  through  the  point  (1,3). 

2.  Draw  the  curve  y  =  x-  +  2  on  a  large  scale.  Tlirough  the  point 
(1,  3)  draw  secants  which  make  Ax  =  1,  ^,  0.1,  0.01,  respectively.  Calcu- 
late the  slope  of  each  of  these  secants  and  show  that  the  values  are  ap- 
proaching the  value  of  the  slope  of  the  curve  at  (1,  3). 

3.  Find  the  slope  of  the  curve  and  the  equation  of  the  tangent  to 
each  of  the  following  curves  at  the  point  mentioned.  Verify  each  answer 
as  in  Ex.  1. 

(ffl)  j/  =  3x2;  (1,3).  ((?)  2/=x2-t-4x-5;   (1,0) 

(6)  2/ =  2x2- 5;   (2,3).  (e)  y  =  x^  +  x^  ;  (1,2). 

(c)  y  =  x^;  (1,  1).  (/)  y  =  x^-3x  +  4:;  (2,6). 

4.  Find  the  slope  of  the  curve  ?/  =  x2  —  3x+l  at  any  point  x  =  a  ; 
from  this  find  the  highest  (maximum)  or  lowest  (minimum)  point  (if 
any),  and  show  in  what  portions  the  curve  is  rising  or  falling. 

5.  Draw  the  following  curves,  using  for  greater  accuracy  the  precise 
values  of  x  and  y  at  the  highest  (maximum)  and  the  lowest  (minimum) 
points,  and  the  knowledge  of  the  values  of  x  for  which  the  curve  rises  or 
falls.  The  slope  of  the  curve  at  the  point  where  x  =  0  is  also  useful  in 
(6),  (c),  (e),  (g). 

(ffl)   y  =  x2  -I-  5  X  +  2.       {(I)  y  =  x*.  (g)  y  =  2  .r^  -  8  x. 

(6)   y  =  xK  (e)   2/  =  -x2  +  3x.  (h)  y  =  x^  -  C>  x  +  5. 

(c)   2/  =  x3-3x4-4.       (/)  y  =  3  +  12x-x3.      (0   y  =  x'^  +  x-. 

6.  Show  that  the  slope  of  the  graph  of  y  =  ax  +  b  is  always  m  =  a, 
(1)  geometrically,  (2)  by  the  methods  of  §  6. 

7.  Show  that  the  lowest  point  on  y  =  x-  +  px  +  q  is  the  point  where 
z  =  — p/2,  (1)  by  Analytic  Geometry,  (2)  by  the  methods  of  §  0. 

8.  The  normal  to  a  curve  at  a  point  is  defined  in  Analytic  Geometry 


12  DERIVATIVES  [II,  §  6 

to  be  the  perpendicular  to  the  tangent  at  that  point.  Its  slope  n  is  shown 
to  be  the  negative  reciprocal  of  the  slope  m  of  the  tangent  -.  n  =—  \/m. 
Find  the  slope  of  the  normal,  and  the  equation  of  the  normal  in  Ex.  1  •, 
in  each  of  the  equations  under  Ex.  3. 

9.  The  slope  w  of  the  curve  y  =  x^  at  any  point  where  x  =  a  is 
m=2  a.  Show  that  the  slope  is  +1  at  the  point  where  a  =  1/2.  Find  the 
points  where  the  slope  has  the  value  —  1,  2,  10.  Note  that  if  the  curve 
is  drawn  by  taking  different  scales  on  the  two  axes,  the  slope  no  longer 
means  the  tangent  of  the  angle  made  with  the  horizontal  axis. 

10.  Find  the  points  on  the  following  curves  where  the  slope  has  the 
values  assigned  to  it ; 

(a)  ?/=x2-3x  +  6;  (m  =  l,  -1,2). 
(6)  y=r?;  (m  =  0,  +  1,  +  6). 
(c)  2/  =  a;3  -  3  X  +  4  ;  (in  =  9,  1) . 

11.  Show  that  the  curve  y  =  x^  — 0.03x  +  2  has  a  minimum  at  (0.1, 
1.998)  and  a  maximum  at  (—  0.1,  2.002).  Draw  the  curve  near  the  point 
(0,  2)  on  a  very  large  scale. 

12.  Draw  each  of  the  following  curves  on  an  appropriate  scale  ;  in 
each  case  show  that  the  peculiar  twist  of  the  curve  through  its  maximum 
and  minimum  would  have  been  overlooked  in  ordinary  plotting  by 
points : 

(ffl)  2/  =48x3-x  +  l• 
[HINT.     Use  a  very  small  vertical  scale  and  a  rather  large  horizontal 
scale.     The  slope  at  x  =  0  is  also  useful.] 

(6)   2/  =  x3^30x2  +  297x. 

[Hint.  Use  an  exceedingly  small  vertical  scale  and  a  moderate  hori- 
zontal scale.    The  slope  at  x  =  10  is  also  useful.] 

7.  Speed.  An  important  case  of  rate  of  change  of  a  quan- 
tity is  the  rate  at  which  a  body  moves,  —  its  speed. 

Consider  the  motion  of  a  body  falling  from  rest  under  the 
influence  of  gravity.  During  the  first  second  it  passes  over 
16  ft.,  during  the  next  it  passes  over  48  ft.,  during  the  third 
over  80  ft.  In  general,  if  t  is  the  number  of  seconds,  and  s 
the  entire  distance  it  has  fallen,  s  =  16  «^  if  the  gravitational 
constant  g  be  taken  as  32.  The  graph  of  this  equation  (see 
Fig.  6)  is  a  parabola  with  its  vertex  at  the  origin. 


'm^' 


II,  §  7] 


SPEED 


13 


The  speed,  that  is  the  rate  of  increase  of  the  space  passed 
over,  is  the  slope  of  this  curve,  i.e. 
liin  As/ At. 

At=0 

This  may  be  seen  directly  in  another  way.  The  average 
speed  for  an  interval  of  time  At  is  found  by  dividing  the  dif- 
ference between  the  space  passed  over  at  the  beginning  and  at 
the  end  of  that  interval  of  time  by  the  difference  in  time :  i.e. 
the  average  speed  is  the  difference  quotient  As  -i-  At.  By  the 
speed  at  a  given  instant  we  mean  the  limit  of  the  average  speed 
over  an  interval  A^  beginning  or  ending  at  that  instant  as  that 
interval  approaches  zero,  i.e. 

speed  =  lim  As/ At. 


Taking  the  equation  s  =  16  t-,  if  <  =  1/2,  s 
After  a   lapse   of   time   At,  the 
new  values  are  t  =  1/2  -|-  At,  and 
s  =  16(l/2  +  A02(«inFig.  6). 

Then 


4  (see  point  P  in  Fig.  6). 


As 


16(1/2 +  A02- 4 
16  At  +  16  M^, 


As/ At  =  16  +  16  At. 
Whence 
speed  —  lim  — 

A(^  A« 

=  lim  (16  +  16A0=16; 

Af=0 

that  is,  the  speed  at  the  end  of 
the  first  half  second  is  16  ft.  per 
second. 

Likewise,  for  any  value  of  t, 
SAy  t=  T,  s=  16  r-  ;  while  for 
t  ^T  +  At,  s  =  16  (r+  At)-; 
hence 

,      As      16(r- 
average  speed  =  —  =  — ^^ — 


4-           ^           /- 

4^                     t 

V                   -1 

X                   4- 

\                   I 

J                        64                              C 

\                                                 S:  =  16/-i/ 

V              J^ 

A              t 

t               U^ 

\                       M 

1              >^Jjr  A/  [       , 

1           iw    ■' 

Aty 


32  r  +  16  At 


and 


speed  ■■ 


:  lim  —  =  32  T. 

Al=^  At 


14 


DERIVATIVES 


[II,  §  8 


Thus,  at  the  end  of  two  seconds,  T  =  2,  and  the  speed  is  32  ■  2  =  64,  in 
feet  per  second. 

8.  Component  Speeds.  Any  curve  may  be  regarded  as  the 
path  of  a  moving  point.  If  a  point  P  does  move  along  a  curve, 
both  X  and  y  are  fixed  when  the  time  t  is  fixed.  To  specify 
the  motion  completely,  we  need  equations  which  give  the  values 
of  X  and  ?/  in  terms  of  t. 

The  horizontal  speed  is  the  rate  of  increase  of  x  with  respect 
to  the  time.  This  may  be  thought  of  as  the  speed  of  the  pro- 
jection iW  of  P  on  the  a;-axis.  As  shown  ih  §  7,  this  speed  is 
the  limit  of  the  difference  quotient  A.t;  -i-  A^  as  A^  =  0. 

Likewise  the  vertical  speed  is  the  limit  of  the  difference  quo- 
tient   Ay -7- At   as   M=0. 

/' 


y 


Since  the  slope  m  of  the 
curve  P  is  the  limit  of 
Ay  -H  Ax  as  Ax  =  0 ;  and 
since 

Ay  _  Ay  .  Ax 
Ax~~At  '   Ai' 

it  follows  that 
m  =  (vertical  speed)  -¥■  (horizontal  spieed) ; 

that  is,  the  slope  of  the  curve  is  the  ratio  of  the  rate  of  increase 
of  y  to  the  rate  of  increase  of  x. 


Fig. 


9.  Continuous  Functions.  In  §§  4-8,  we  have  supposed  that 
the  curves  used  were  smooth.  The  functions  which  we  have 
had  have  all  been  representable  by  smooth  curves ;  except 
perhaps  at  isolated  points,  to  a  small  change  in  the  value  of 
one  coordinate,  there  has  been  a  correspondingly  small  change 
in  the  value  of  the  other  coordinate.  Throughout  this  text, 
unless  the  contrary  is  expressly  stated,  the  functions  dealt  with 
will  be  of  the  same  sort.  Such  functions  are  called  continuous. 
(See  §  10,  p.  17.) 


II,  §  9]  SPEED  15 

The  curve  y=l/x  is  continuous  except  at  the  point  a;  =  0  ;  y  =  tan  x 
is  continuous  except  at  the  points  x  =  ±  7r/2,  ±  3  ir/2,  etc.  Such  excep- 
tional points  occur  frequently  ;  we  do  not  discard  a  curve  because  of  them, 
but  it  is  understood  that  any  of  our  results  may  fail  at  such  points. 

EXERCISES  IV.  — SPEED 

1.  From  the  formula  s  =  16  t-,  calculate  the  values  of  s  when  f  =  1,  2, 
1.1,  1.01,  1.001.  From  these  values  calculate  the  average  speed  between 
t  =  \  and  t  =  2;  between  t  —  1  and  t—l.l;  between  t  =  \  and  t  =  1.01  ; 

■  between  t  =  1  and  t  =  1.001.     Show  that  these  average  speeds  are  succes- 
sively nearer  to  the  speed  at  the  instant  <  =  1. 

2.  Calculate  as  in  Ex.  1  the  average  speed  for  smaller  and  smaller  in- 
tervals of  time  after  t  =2;  and  show  that  these  approach  the  .speed  at  the 
instant  t  =  2. 

3.  A  body  thrown  vertically  downwards  from  any  height  with  an 
original  velocity  of  100  ft.  per  second,  passes  over  in  time  t  (in  seconds)  a 
distance  s  (in  feet)  given  by  the  equation  s  —  100  t  +  \Q  (^  (if  gr  =  32,  as 
in  §  7).  Find  the  speed  v  at  the  time  t  =  \  ;  at  the  time  t  —  2;  at  the 
time  t  =  i;  at  the  time  t  =  T. 

4.  In  Ex.  3  calculate  the  average  speeds  for  smaller  and  smaller  in- 
tervals of  time  after  (  =  0  ;  and  show  that  they  approach  the  original 
speed  vo  =  100.     Repeat  the  calculations  for  intervals  beginning  with  t=2. 

5.  Calculate  the  speed  of  a  body  at  the  times  indicated  in  the  follow- 
ing possible  relations  between  s  and  t : 

(a)  s  =  «2;  «  =  1,  2,  10,    T.  (c)   s  = -16  «2  +  160  < ;  <  =  0,  2,  5. 

(6)  s  =  16f^-100t;  t  =  0,2,  T.     (d)  s  =  «3-3«  +  4;  «  =  0,   1/2,  1. 

6.  The  relation  (c)  in  Ex.  5  holds  (approximately,  since  gr  =  32  ap- 
proximately) for  a  body  thrown  upward  with  an  initial  speed  of  160  ft. 
per  second,  where  s  means  the  distance  from  the  starting  point  counted 
positive  upwards.  Draw  a  graph  which  represents  this  relation  between 
the  values  of  s  and  t. 

In  this  graph  mark  the  greatest  value  of  s.  What  is  the  value  of  v  at 
that  point  ?    Find  exact  values  of  s  and  t  for  this  point. 

7.  A  body  thrown  horizontally  with  an  original  speed  of  4  ft.  per 
second  falls  in  a  vertical  plane  curved  path  so  that  the  values  of  its  hori- 
zontal and  its  vertical  distances  from  its  original  position  are  respectively, 
X  =  4  «,  1/  =  16  (-,  where  y  is  measured  downwards.  Show  that  the  vertical 
speed  is  32  T,  and  that  the  horizontal  .speed  is  4,  at  the  instant  t  =  T. 
Eliminate  t  to  show  that  the  path  is  the  curve  y  =  z*. 


16  DERIVATIVES  [II,  §  10 

8.  Show  by  Ex.  7  and  §  8  that  the  slope  of  the  curve  ?/  =  a;^  at  the 
point  where  t  =  l,  i.e.  (-i,  10),  is  32  ~  4,  or  8.  Write  the  equation  of  the 
tangent  at  that  point. 

9.  Show  that  the  slope  of  the  curve  y  =  x^  (Ex.  7)  at  the  point  (a,  a^), 
i.e.  t  =  a/4,  is  2  a,  from  Ex.  7  and  §  8  ;  and  also  directly  by  means  of  §  6. 

10.  If  a  body  moves  so  that  its  horizontal  and  its  vertical  distances 
from  the  starting  point  are,  respectively,  x  =  16  t^,  y  =  4  t,  show  that  its 
path  is  the  curve  y'^  =  x  ;  that  its  horizontal  speed  and  its  vertical  speed 
are,  respectively,  32  T  and  4,  at  the  instant  t  =  T. 

11.  From  Ex.  10  and  §  8  show  that  the  slope  of  the  curve  y^  ^x&t  the 
point  (16,  4),  i.e.  when  f  =  1,  is  4  h-  32  =  1/8.  Write  the  equation  of  the 
tangent  at  that  point. 

12.  From  Ex.  10  and  §  8  show  that  the  slope  of  the  curve  y"^  =  x  aX  the 
point  where  t=  T  is  4  --  (32  T)  =  1/(8  T)  =  1/(2  A:),  where  k  is  the  value 
of  y  at  the  point.     Compare  this  result  with  that  of  Ex.  8. 

10.  Limits.  Infinitesimals.  We  have  been  led  in  what  pre- 
cedes to  make  use  of  limits.  Thus  the  tangent  to  a  curve  at 
the  point  P  is  defined  by  saying  that  its  slope  is  the  limit  of 
the  slope  of  a  variable  secant  through  P;  the  speed  at  a  given 
instant  is  the  limit  of  the  average  speed ;  the  difference  of  the 
two  values  of  x,  Ax,  was  thought  of  as  approacldng  zero ;  and 
so  on.  To  make  these  concepts  clear,  the  following  precise 
statements  are  necessary  and  desirable. 

When  the  difference  betiveen  a  variable  x  and  a  constant  a  he- 
comes  and  remains  less,  in  absolute  value,*  than  ani/  preassigyied 
positive  quantity,  however  small,  then  a  is  the  limit  of  the  vari- 
able x. 

We  also  use  the  expression  "x  approaches  a  as  a  limit,"  or, 
more  simply,  "  x  approaches  o."  The  symbol  for  limit  is  Urn ; 
the  symbol  for  approaches  is  =  :  thus  we  may  write  lim  x==a, 
or  x  =  a,  or  lim  (a  —  x)  =  0,  or  a  —  x=0. 

When  the  limit  of  a  variable  is  zei'o,  the  variable  is  called 

*  When  dealing  with  real  numbers,  absolute  value  is  the  value  without 
regard  to  signs  so  that  the  absolute  value  of  —  2  is  2.  A  convenient  symbol 
for  it  is  two  vertical  lines ;  thus  |3  —  7 1=  4. 


II,  §  11]  LIMITS  17 

an  infinitesimal.  Thus  a  —  x  above  is  an  infinitesimal.  The 
difference  between  any  variable  and  its  limit  is  always  an  in- 
finitesimal. When  a  variable  x  approaches  a  limit  a,  an;/  con- 
tinuous function  J\x)  approaches  the  limit  /(a):  thus,  if  y=f{x) 
and  b  =f{a),  we  may  write 

lim  y  =  b,  or  lim/(a;)  =f(a). 

This  condition  is  the  precise  definition  of  continuity  at  the 
point  x  =  a.     (See  §  0,  p.  14.) 

11.  Properties  of  Limits.  The  following  properties  of  limits 
will  be  assumed  as  self-evident;  some  of  them  have  already 
been  used  in  the  articles  noted  below. 

Theorem  A.  The  limit  of  the  siim.  of  two  variables  is  the  sum 
of  the  limits  of  the  ttco  variables.  This  is  easily  extended  to  the 
case  of  more  than  two  variables.     (Used  in  §§4,  6,  and  7.) 

Theorem  B.  The  limit  of  the  product  of  two  variables  is  the 
product  of  the  limits  of  the  variables.     (Used  in  §§4,  6,  and  7.) 

Theorem  C.  The  limit  of  the  quotient  of  one  variable  divided 
by  another  is  the  quotient  of  the  limits  of  the  variables,  provided 
the  limit  of  the  divisor  is  not  zero.     (Used  in  §  8.) 

The  exceptional  case  in  Theorem  C  is  really  the  most  in- 
teresting and  important  case  of  all.  The  exception  arises 
because  when  zero  occui-s  as  a  denominator,  the  division  can- 
not be  performed.  In  finding  the  slope  of  a  curve,  we  consider 
lim  (Ay/A-r)  as  A.r  approaches  zero;  notice  that  this  is  pre- 
cisely the  case  ruled  out  in  Theorem  C.  Again,  the  speed  is 
lim.(As/A^)  as  M  approaches  zero.  The  limit  of  any  such 
difference  quotient  is  one  of  these  exceptional  cases. 

Now  it  is  clear  that  the  slope  of  a  curve  (or  the  speed  of  an 
object)  may  have  a  great  variety  of  values  in  different  cases : 
no  one  answer  is  sufficient  for  all  examples,  in  the  case  of  the 
limit  of  a  quotient  when  the  denominator  approaches  zero. 


18  DERIVATIVES  [II,  §  12 

Theorem  D.  Tlie  limit  of  the  ratio  of  two  infinitesimals  de- 
pends upon  the  law  connecting  them;  otherwise  it  is  quite  inde- 
terminate. Of  this  the  student  will  see  many  instances ;  for 
the  Differential  Calculus  consists  of  the  consideration  of  just  such 
limits.  In  fact,  the  very  reason  for  the  existence  of  the  Diifer- 
ential  Calculus  is  that  the  exceptional  case  of  Theorem  C  is 
important,  and  cannot  be  settled  in  an  offhand  manner. 

The  thing  to  be  noted  here  is,  that,  no  matter  how  small  two 
quantities  may  be,  their  ratio  may  be  either  small  or  large ; 
and  that,  if  the  two  quantities  are  variables  whose  limit  is 
zero,  the  limit  of  their  ratio  may  be  either  finite,  zero,  or 
non-existent.  In  our  work  with  such  forms  we  shall  try  to 
substitute  an  equivalent  form  whose  limit  can  be  found. 
Obviously,  to  say  that  two  variables  are  vanishing  implies 
nothing  about  the  limit  of  their  ratio. 

12.  Ratio  of  an  Arc  to  its  Chord.  Another  important  illus- 
tration of  a  ratio  of  infinitesimals  is  the  ratio  of  the  chord  of 

a  curve  to  its  subtended  arc :  ,      ,  „  „ 

p  _  chord  PQ 

~    arcPQ 

If  Q  approaches  P,  both  the  arc 
and  the  chord  approach  zero.  At 
any  stage  of  the  process  the  arc  is 
greater  than  the  chord ;  but  as  Q 
approaches  P  this  difference  di- 
minishes very  rapidly,  and  the 
ratio  R  approaches  1 : 

V      „      T      chord  PQ      ^ 

hm  R  =  lim  --^  =  1. 

<i=p         pQ=o   arc  PQ 

This  property  is  self-evident  because  it  amounts  to  the  same 
thing  as  the  definition  of  the  length  of  the  curve ;  we  ordinarily 
think  of  the  length  of  an  arc  of  a  curve  as  the  limit  of  the 
length   of   an   inscribed   broken   line,  as  the  lengths  of   the 


II,  §  13]  LIMITS  19 

segments  of  the  broken  line  approach  zero.  Thus,  the  length 
of  circumference  of  a  circle  is  defined  to  be  the  limit  of  the 
perimeter  of  an  inscribed  polygon 
as  the  lengths  of  all  its  sides  ap- 
proach zero.  This  would  not  be 
true  if  the  ratio  of  an  arc  to  its 
chord  did  not  approach  1.* 


13.  Ratio  of  the  Sine  of  an 
Angle  to  the  Angle.  In  a  circle, 
the  arc  PQ  and  the  chord  PQ  can 
be  expressed  in  terms  of  the  angle 
at  the  center.     Let  a  =  Z  QOP/2 ;  ^^^-  •'• 

then  arc  PQ  =  2  a  x  r  if  a  is  measured  in  circular  measure  (see 
Tables,  II,  F,  3) ;  and  the  chord  PQ  =  2  r  sin  «,  since  r  sin  a  = 
AP. 

It  follows  that 

1  •  _  chord  PQ      ,.      2  r  sin  a      ,.      sin  «      ^ 

lim  ^=:lim  =  lim  =  1; 

a=o     arc  PQ        a=o      2  ra  a=o      a 

hence  lim  ^^^  =  1, 

0  =  0         « 

for  we  have  just  seen  that  the  limit  of  the  ratio  of  an  infini- 
tesimal chord  to  its  arc  is  1. 

This  result  is  very  important  in  later  work ;  just  here  it 
serves  as  a  new  illustration  of  the  ratio  of  infinitesimals :  the 
ratio  of  the  sine  of  an  angle  to  the  angle  itself  (measured  in  cir- 
cular measure)  approaches  1  as  the  angle  approaches  zero. 

14.  Infinity.  Theorem  D  accounts  for  the  case  when  the 
numerator  as  well  as  the  denominator  in  Theorem  C  is  infini- 
tesimal.    There  remains  the  case  when  the  denominator  only 

*This  point  of  view  is  fundamental.  See  Gonrsat-Hedrick,  Mathematical 
Analysis,  Vol.  I,  §80,  p.  161.  At  some  exceptional  points  the  property  may 
fail,  but  such  points  we  always  subject  to  special  investigation. 


20  DERIVATIVES  [II,  §  14 

is  infinitesimal.  A  variable  whose  reciprocal  is  infinitesimal  is 
said  to  become  infinite  as  the  reciprocal  approaches  zero. 

Thus  y  —  1/x  is  a  variable  whose  reciprocal  is  x.  As  x  ap- 
proaches zero,  y  is  said  to  become  infinite.  Notice  however 
that  y  has  no  value  whatever  when  x  =  0. 

Likewise  y  =  sec  a;  is  a  variable  whose  reciprocal,  cos  x,  is 
infinitesimal  as  x  approaches  7r/2 ;  hence  we  say  that  sec  x  be- 
comes infinite  as  x  approaches  7r/2. 

In  any  case,  it  is  clear  that  a  variable  which  becomes  infinite 
becomes  and  remains  larger  in  absolute  value  than  any  pre- 
assigned  positive  number,  however  large. 

The  student  should  carefully  notice  that  infinity  is  not  a 
number ;  when  we  say  that  "  sec  x  becomes  infinite  as  x  ap- 
proaches 7r/2,"  *  we  do  not  mean  that  sec  (7r/2)  has  a  value,  we 
merely  tell  what  occurs  when  x  approaches  7r/2. 

EXERCISES   v.  — LIMITS   AND   INFINITESIMALS 

1.  Imagine  a  point  traversing  a  line-segment  in  such  fashion  that  it 
traverses  lialf  the  segment  in  the  first  second,  half  the  remainder  in  the 
next  second,  and  so  on  ;  always, half  the  remainder  in  the  next  following 
second.  Will  it  ever  traverse  the  entire  line  ?  Show  that  the  remainder 
after  t  seconds  is  1/2',  if  the  total  length  of  the  segment  is  1.  Is  this 
infinitesmal  ?     Why  ? 

2.  Show  that  the  distance  traversed  by  the  point  in  Ex.  1  in  t  seconds 
is  1/2  -H  1/2-  +  •••  +  1/2'.  Show  that  this  sum  is  equal  to  1  —  1/2' ;  hence 
show  that  its  limit  is  1.  Show  that  in  any  case  the  limit  of  the  distance 
traversed  is  the  total  distance,  as  t  increases  indefinitely. 

3.  Show  that  the  limit  of  3  —  x^  as  x  approaches  zero  is  3.  State  this 
result  in  the  symbols  used  in  §  10.  Draw  the  graph  of  y  =  S  —  x^  and , 
show  that  y  approaches  3  as  x  approaches  zero. 

4.  Evaluate  the  following  limits  : 

(a)    lim(2-5x  +  3x2).    (d)    lim|^|^-      (g) 


4  +  2x2        ^  "    x=TX^+2x  +  3 
*  Or,  as  is  stated  iu  short  form  in  many  texts,  "sec  (t/2)  =  oo  ." 


11,  §  14] 


LIMITS 


21 


5.  If  the  numerator  and  denominator  of  a  fraction  contain  a  common 
factor,  that  factor  may  be  canceled  in  finding  a  limit,  since  the  value  of 
the  fraction  which  we  use  is  not  changed.  Evaluate  before  and  after 
canceling  a  common  factor  : 

(x  +  2)(x+l)  ,^,    ,■•„       x(x  +  2) 

(2x  +  3)(x+l) 


(«) 


(6)    lim 


=o(x  +  l)(x  +  2) 
Evaluate  after  (not  before)  removing  a  common  factor  : 
(c) 


lim-- 

z  =  0  X 

(.)   H.-^a^. 

(0    lim    ^"-'l^^-'l 
x^i(2x  +  3)(x-l) 

lira^^-/ 

x  =  l     X-  I 

(">   -mf  ■ 

,,  -      ,.        X"           (0,    71  >1, 

(h)    lim-=  -     '      ^    ' 
x-o  X        (  1,  n  =  1. 

Show  that 

lira      2x2  +  3 

=  2. 

x=aoa;2  +  43;  -I-  5 

[Hint.    Divide  numerator  and  denominator  by  x"^ ;  then  such  terms  as  S/x* 
approach  zero  as  x  becomes  infinite.] 


7.   Evaluate  : 
2x  +  1 
3x  +  2' 


(a)    lim 


!x2-4 


(d)    lim 


VI 


ax+b 
mx  +  n 

Vax-  +  bx  4- 

c 

Vx2  -  1 


8.  Let  O  be  the  center  of  a  circle  of  radius  r  —  OB,  and  let  a  =  Z  COB 
be  an  angle  at  the  center.  Let  BT  be  perpen- 
dicular to  OB,  and  let  BF  be  perpendicular  to 
OC.  Show  that  OF  approaches  OC  as  a  ap- 
proaches zero  ;  likewise  arc  CB  =  0,  arc  DB=:  0, 
and  FC  =  0,  as  a  =  0. 

9.  In  the  figure  of  Ex.  8,  show  that  the  ob- 
vious geometric  inequality  FB <_  arc  CB<BT 
is  equivalent  to  r  sin  a<r  ■  a-^r  tan  «,  if  «  is  measured  in  circular 
measure.  Hence  show  that  a/sin  a  lies  between  1  and  1/cos  a,  and  there- 
fore that  lim  («/sin  a)  =  1  as  a  =  0.     (Verification  of  §  13.) 

10.    In  the  figure  of  Ex.  8,  show  that 


lim 


FB 


lira 
Oil)   r 


OF 


BT 


0:    lim 


FC 


0 


arc  CB 


=  0. 


22  DERIVATIVES  [II,  §  15 

11.  Show  that  the  following  quantities  become  infinite  as  the  independ- 
ent variable  approaches  the  value  specified  ;  in  (a)  and  (6)  draw  the  graph. 

(a)    lim--  (c)    lira --^,  (Ex.  8).         (e)    lim^,(«<l). 

a!  =  0  X'^  "=0  r  O  1  =  0    X 

(6)    lim^-  (d)    lim -^,  (Ex.8).        (/)    lim— 1^+^— . 

12.  As  the  chord  of  a  circle  approaches  zero,  which  of  the  following 
ratios  has  a  finite  limit,  which  is  infinitesimal,  and  which  is  becoming 
infinite  :  the  chord  to  its  arc  ;  the  radius  to  the  chord  ;  the  sector  of  the 
arc  to  the  triangle  cut  off  by  the  chord  ;  the  area  of  the  circle  to  the  sector  ; 
the  chord  of  twice  the  arc  to  the  chord  of  thrice  the  arc  ;  the  radius  of  the 
circle  to  the  chord  of  an  arc  a  thousand  times  the  given  arc  ? 

13.  Is  the  sum  of  two  infinitesimals  itself  infinitesimal  ?  Is  the  dif- 
ference ?  Is  the  product  ?  Is  the  quotient  ?  Is  a  constant  times  an 
infinitesimal  an  infinitesimal  ? 

14.  If  to  each  of  two  integers  an  infinitesimal  is  added,  show  that  the 
ratio  of  these  sums  differs  from  that  of  the  integers  by  an  infinitesimal. 
[See  Ex.  4  (h).'] 

15.  Show  that  the  graph  of  y  =f(x)  has  a  vertical  asymptote  if  f(x) 
becomes  infinite  as  x=a.  Illustrate  this  by  drawing  the  following 
graphs : 

(a)  2/  =  -^^+^.  (c)y  =  -^ (e)y-        ^ 


2  1  —  cos  X  Vl  —  x"'^ 

,  ^^-'     •      W2/  =  ^4^-        (f)y  =  ''^- 

(x  +  1)  (x  —  5)  e*  —  e  '  ex  +  d 


15.  Derivatives.  While  such  illustrations  as  those  in  §  12 
and  Exercises  V,  above,  are  interesting  and  reasonably  impor- 
tant, by  far  the  most  important  cases  of  the  ratio  of  two  infini- 
tesimals are  those  of  the  type  studied  in  §§  4-8,  in  which  each 
of  the  infinitesimals  is  the  difference  of  two  values  of  a  varia- 
ble, such  as  Ay/Ax  or  As/At.  Such  a  difference  quotient 
Ay/ Ax  of  y  with  respect  to  x  evidently  represents  the  average 
rate  of  increase  of  y  with  respect  to  x  in  the  interval  Ax ;  if  x 
represents  time  and  y  distance,  then  Ay/ Ax  is  the  average 
speed  over  the  interval  Ax  (§  7,  p.  13);  if  y  =if(x)  is  thought 


II,  §  16]  FORMAL   DIFFERENTIATION  23 

of  as  a  curve,  then  Ay/^x  is  the  slope  of  a  secant  or  the  aver- 
age rate  of  rise  of  the  curve  in  the  interval  Ax  (§  4,  ]).  6). 

The  limit  obtained  in  such  cases  represents  the  instantaneous 
-''ate  of  increase  of  one  variable  with  respect  to  the  otlier,  — 
this  may  be  the  slope  of  a  curve,  or  the  speed  of  a  moving 
object,  or  some  other  rate,  depending  upon  the  nature  of  the 
problem  in  which  it  arises. 

In  general,  the  limit  of  the  quotient  A/y/A.c  of  two  infinitesimal 
differences  is  called  the  derivative  of  y  with  respect  to  a? ;  it  is 
represented  by  the  symbol  dij/dx  : 

-^  =  derivative  of  y  witli  respect  to  x  =  lim  — ^• 
dx  Ax=oAx 

Henceforth  we  shall  use  this  new  symbol  dy/dx  or  other 
convenient  abbreviations;  *  but  the  student  must  not  forget  the 
real  meaning :  slojje,  in  the  case  of  curve ;  speed,  in  the  case  of 
motion;  some  other  tangible  concept  in  any  new  problem 
which  we  msiy  undertake ;  in  every  case  the  rate  of  increase  of 
y  ivith  respect  to  x. 

Any  mathematical  formulas  we  obtain  will  aj^ply  in  any  of 
these  cases ;  we  shall  use  the  letters  x  and  ?/,  the  letters  s  and 
t,  and  other  suggestive  combinations ;  but  the  student  should 
remember  that  any  formula  written  in  x  and  y  also  holds  true, 
for  example,  with  the  letters  s  and  t,  or  for  any  other  pair  of 
letters. 

16.   Formula  for  Derivatives.     If  we  are  to  find  the  value  of 
\  a  derivative,  as  in  §§  4-7,  we  must  have  given  one  of  the  vari- 
ables ?/  as  a  function  of  the  other  x : 

(1)  y=f{^)- 

If  we  think  of  (1)  as  a  curve,  we  may,  as  in  §  4,  take  any 

*  Often  read  "  the  x  derivative  of  y."  Other  names  sometimes  used  are 
differential  coefficient,  and  derived  function.  Other  convenient  notations 
often  used  are  Dzy,  ?/x,  ?/'.  y  ;  tlie  last  two  are  not  safe  unless  it  is  otherwise 
clear  what  the  independent  variable  is. 


24  DERIVATIVES  [II,  |  16 

point  P  whose  coordinates  are  x  and  y,  and  join  it  by  a  secant 
PQ  to  any  otlier  point  Q,  whose  coordinates  are  x  +  Aa;,  y  +  Ay. 
Here  x  and  y  represent  fixed  values 
of  X  and  y ;  this  will  prove  more  con- 
venient than  to  use  new  letters 
each  time,  as  we  did  in  §§  4-7. 

Since  P  lies  on  the  curve  (1),  its 
coordinates  (x,  y)  satisfy  the  equa- 
tion (1),  y  =f(x).     Since  Q  lies  on 
(1),  ar-f  Aa;  and  y  +  ^y  satisfy  the 
same  equation ;  hence  we  must  have 

(2)  y  +  £^y=f{x  +  ^x). 
Subtracting  (1)  from  (2)  we  get 

(3)  ^y=f{x  +  ^xr-f{x)', 
whence  the  difference  quotient  is 

(4)  ^  ^/(a;-f-A.T)-/(a;)  ^  ^^^^         ^i  p^ 
^  ^      Ax                  Ax                            J        ^  > 

and  therefore  the  derivative  is 

(5)  4?^  =  lim  fl^  ^  lim  /(•^  +  A.r)-/(^)  ^  ^j        ^^  p^ 

This  formula  is  often  convenient;  we  shall  apply  it  at  once. 

17.  Rule  for  Differentiation.  The  process  of  finding  a  de- 
rivative is  called  differentiation.    To  apply  formula  (5)  of  §  16 : 

(A)  Find  {y  +  Ay)  by  substitzUing  (x  +  Ax)  for  x  in  the  given 
function  or  equation;  this  gives  y  +  Ay  =f(x -f  Ax*).  > 

(B)  Subtract  y  froin  y-\- Ay ;  this  gives  Ay =/(£c-fAa;)—/(a;). 

(C)  Divide  Ay  by  Ax  to  find  the  difference  quotient  Ay /Ax; 
simplify  this  result. 

(D)  Find  the  limit  of  Ay /Ax  as  Ax  approaches  zero ;  this  I 
result  is  the  derivative,  dy/dx. 

*  Instead  of  slope,  read  speed  in  case  the  problem  deals  with  a  motion,  as 
in  §  7.  In  general,  Ajz/As;  is  the  average  rate  of  increase,  and  dy/dx  is  the  i 
instantaneous  rate. 


II,  §  17]  FORMAL   DIFFERENTIATION  25 

Example  1.     Given  y  =f(x)=x'^,  to  find  dy/dx. 

(A)  f{x  +  Ax)  =  {x+Axy^. 

(B)  Ay  =f{x  +  Ax)-/(x)  =  (x  +  AxY  -  x^  =  2xAx  +  A^'. 

(C)  Ay/Ax  =  (2 xAx  +  Ax")  H- Ax  =  2  X  +  Aa:. 
(Z>)             dy/dx=  lim  Ay/Ax=  lim  (2x  +  Ax)  =  2x. 

Ax  =  0  Ai  =  0 

Compare  this  work  and  the  answer  with  the  work  of  §  4,  p.  6. 

Example  2.     Given  y  =/(x)  =x3  —  12x  +  7,  to  find  dy/dx. 
{A)  /(x  + Ax)  =  (x+ Ax)3-12(x  + Ax)+7. 

{B)      Ai/=/(x  + Ax)-/(x)  =  3x2Ax  +  3xA?  +  Ax^-  12Ax. 
(  C)  Ay /Ax  =  3  x2  +  3  xAx  +  Ax-  -  12. 

(Z>)       dy/dx  =  lim  Ay/ Ax  =  lim  (3  x'-  +  3  xAx  +  Ax"  -  12)  =  3 x2  - 12. 

Ai  =  0  Ai  =  0 

Compare  this  work  and  the  answer  with  the  work  of  Example  3,  §  6. 
Example  3.     Given  y  =/(x)  =  l/x^,  to  find  dy/dx. 

{A)  /(x+Ax)=         1 

{B)        Ay=/(x  +  Ax)-/(x)= 1 1  ^  _  2xAx  +  A^ 

(C)  Ay/ Ax 

^  ^'  x2(a;  +  Ax)2 

(2))        d2,/dx=  lim^=  lim  T  - -1^+-4^1  =  -  2^  =  -  1 . 
Ax=oAx      Ai=oL      x2(x  +  Ax)2J  X*  X* 

Example  4.     Given  y  =/(x)=  Vx,  to  find  dy/dx,  or  df{x)/dx. 


(X 

+  Ax)2 

1 

1 

(X 

+  Ax)2 

x2 

2x  + Ax 

(^4)  /(x  +  Ax)  =  Vx  +  Ax. 


1^  (B)        Ay  =  /(x  +  Ax)  -  /(x)  =  Vx  +  Ax  -  Vx. 

(C)  Ay  _  Vx  +  Ax  —  Vx  _  Vx  +  Ax  -  Vx  ^   Vx  +  Ax  +  Vg 

Ax  Ax  ~  Ax  VxTAx  +  Vx 

1 


Vx  +  Ax  +  Vx 

(m  ^=  lim^=  lim- 1 =  -i-. 

''  dx      Ax=oAx     Ax=oVx  + Ax  +  Vx      2Vx 

(Compare  Ex.  11,  p.  16.) 


26  RATES  [II.  §  17 

Example  5.     Given  y  =  /(x)  =  .r^  to  find  df{x)/dx.  _^ 

(^)       /(x  +  Ax)  =  {X  +  Ax)^  =  x^  +  7  x^Ax  +  (terms  with  a  factor  Ax  ) . 
(5)        Ay  =  /(x  +  Ax)  -  /(x)  =  7  x6Ax  +  (terms  with  a  factor  Ax'). 

(C)  Ay/Ax=:  7x6+  (terms  with  a  factor  Ax). 

(D)  dy/dx  =  lira  A2//Ax=  lira  [7  x^  +  (terms  with  a  factor  Ax)]  =  7  x^. 

EXERCISES  VI. -FORMAL  DIFFERENTIATION 
1    Find  the  derivative  of  y  =  x^  with  respect  to  x.     [Corapare  Ex  3 

(c),  p.  11.]     Write  the  equation  of  the  tangent  at  the  pomt  (2,  8)  to  the 

curve  y  =  x*. 

2.   Find  the  derivatives  of  the  following  functions  with  respect  to  x  : 

(a)x2-3x  +  4.  (6)x3-6x  +  7.  (c)   x^  +  5. 

(d)  x^  +  3x^-2.  ie)    x3  +  2x;^-4.  (/)  x^-3x3+5x. 

(.)  h.-  ^'^  x-TT-  ^"^   ^^' 

2x+3 


:-2 


3.  Find  the  equation  of  the  tangent  and  the  equation  of  the  normal 
to  the  curve  y  =  1/x  at  the  point  where  x  =  2.     (See  Ex.  8,  p.  11. ) 

4  Find  the  values  of  x  for  which  the  curve  y  =  x^-\bx  +  1  rises 
and  those  for  which  it  falls  ;  find  the  highest  point  (maximum)  and  the 
lowest  point  (minimum).     Draw  the  graph  accurately. 

5  Draw  accurate  graphs  for  the  following  curves  : 

(a)  2/  =  x3-18x  +  3.  (c)  y  =  x*-32x. 

(ft)y^x3  +  3x^.  (d)y  =  x4-18x2. 

6.    Determine  the  speed  of  a  body  which  moves  so  that 

s=  16«2  + 10«  +  5.  N 

FA  body  thrown  down  from  a  height  with  initial  speed  10  ft.  per  sec- 
ond moves  in  this  way  approximately,  if  .  is  measured  downward  from  a 
mark  5  ft.  above  the  starting  point.] 

7  If  a  body  moves  so  that  its  horizontal  and  its  vertical  distances 
from  a  point  are,  respectively,  .  =.  10  ^  y  =  -  16  i^  +  10  e,  find  xts  hori- 
zontal speed  and  its  vertical  speed.    Show  that  the  path  is 

2/  =  -  16  xVlOO  +  X' 
and  that  the  slope  of  this  path  is  the  ratio  of  the  vertical  speed  to  th« 


II,  §  17]  FORMAL  DIFFERENTIATION  27 

horizontal  speed.  [These  equations  represent,  approximately,  the  motion 
of  an  object  thrown  upward  at  an  angle  of  45°  with  a  speed  10  V2.] 

8.  A  stone  is  dropped  into  still  water.  The  circumference  c  of  the 
growing  circular  waves  thus  made,  as  a  function  of  the  radius  ?•,  isc  =  2  irr. 

Show  that  dcldr  =  2  ir,  i.e.  that  the  circumference  changes  2  ir  times  as 
fast  as  the  radius. 

Let  ^1  be  the  area  of  the  circle.  Show  that  dAjdr  ^'l-irr  \  i.e.  the 
rate  at  which  the  area  is  changing  compared  to  the  radius  is  numerically 
equal  to  the  circumference. 

9.  Determine  the  rates  of  change  of  the  following  variables : 

(a)  The  surface  of  a  sphere  compared  with  its  radius,  as  the  sphere 
expands. 

(6)  The  volume  of  a  cube  compared  with  its  edge,  as  the  cube  enlarges. 

(c)  The  volume  of  a  right  circular  cone  compared  with  the  radius  of 
its  base  (the  height  being  fixed) ,  as  the  base  spreads  out. 

10.  If  a  man  G  ft.  tall  is  at  a  distance  x  from  the  base  of  an  arc  light 
10  ft.  high,  and  if  the  length  of  his  shadow  is  s,  show  that  s/6  =  x/4,  or 
s  =  3x/2.  Find  the  rate  (ds/dx)  at  which  the  length  s  of  his  shadow 
increases  as  compared  with  his  distance  x  from  the  lamp  base. 

11.  The  specific  heat  of  a  substance  (e.g.  water)  is  the  amount  of  heat 
required  to  raise  the  temperature  of  a  unit  volume  of  that  substance  1° 
(Centigrade).  This  amount  is  known  to  change  for  the  same  substance 
for  different  temperatures.  The  average  specific  heat  between  two  tem- 
peratures is  the  ratio  of  the  quantity  of  heat  AH  consumed  in  raising 
the  temperature  divided  by  the  change  At  in  the  temperature  ;  show  that 
the  actual  specific  heat  at  a  given  temperature  is  dH/dt. 

12.  The  coefficient  of  expansion  of  a  solid  substance  is  the  amount  a 
bar  of  that  substance  1  ft.  long  will  expand  when  the  temperature  changes 
1°.  Express  the  average  coefficient  of  expansion,  and  show  that  tlie  coeGB- 
cient  of  expansion  at  any  given  temperature  is  dl/dt,  if  the  bar  is  precisely 
1  ft.  long  at  that  temperature.     (See  also  Ex.  12,  p.  145.) 


CHAPTER  III 

DIFFERENTIATION   OF   ALGEBRAIC   FUNCTIONS 

PART   I.     EXPLICIT   FUNCTIONS 

18.  Classification  of  Functions.  For  convenience  it  is  usual 
to  classify  functions  into  certain  groups. 

A  function  which  can  be  expressed  directly  in  terms  of  the 
independent  variable  x  by  means  of  the  three  elementary 
operations  of  multiplication,  addition,  and  subtraction  is  called 
a  poljmomial  in  x. 

Thus,  ic^(=  cc  •  ic),  2  x^  +  4  a^  —  7  aj  +  3,  ar^  —  4  x-  +  6,  etc.,  are 
polynomials.  The  most  general  polynomial  is  ayX"  +  aiX""^ + 
•••  +  a„_iX  +  a„,  where  the  coefficients  Uq,  Uj,  •••,  a„  are  constants, 
and  the  exponents  are  positive  integers.  Notice  that  raising 
a  quantity  to  a  positive  integral  power  can  be  regarded  as  a 
succession  of  multiplications, 

A  function  which  can  be  expressed  directly  in  terms  of  the 
independent  variable  x  by  means  of  the  four  elementary  opera- 
tions of  multiplication,  division,  addition,  and  subtraction,  is 
called  a  rational  function  of  x.  Thus,  1/x,  (ar*  —  3  x)/{2  x  +  7), 
etc.,  are  rational.  The  most  general  rational  function  is  the 
quotient  of  two  polynomials,  since  more  than  one  division  can 
be  reduced  to  a  single  division  by  the  rules  for  the  combination 
of  fractions.     All  polynomials  are  also  rational  functions. 

If,  besides  the  four  elementary  operations,  a  f miction  re- 
quires for  its  direct  expression  in  the  independent  variable  x 
at  most  the  extraction  of  integral  roots,  it  is  called  a  simple 
algebraic  function  *  of  x.     Thus,  Vx,  ( Vx-^  +  1  —  2)/(3  —  \/x), 

♦Since  the  expression  "algebraic  function"  is  used  in  tlie  broader  sense 
of  §  27  in  advanced  matliematics,  we  sliall  call  these  simple  algebraic 
functions. 

28 


Ill,  §  19]       CLASSIFICATION  OF  FUNCTIONS  29 

etc.,  are  simple  algebraic  functions.  All  rational  functions  are 
also  simple  algebraic  functions. 

Simple  algebraic  functions  which  are  not  rational  are  called 
irrational  functions. 

A  function  which  is  not  an  algebraic  function  is  called  a 
transcendental  function.  Thus,  sin  x,  log  x,  e^,  j^+  tan-^  (1  -f-  x), 
etc.,  are  transcendental. 

In  this  chapter  we  shall  deal  only  with  algebraic  functions. 

19.  Differentiation  of  Polynomials.  We  have  differentiated 
a  number  of  polynomials  in  Chapter  II.  To  simplify  the 
work  to  a  mere  matter  of  routine,  we  need  four  rules : 

Tlie  derivative  of  a  constant  is  zero : 

[I]  1^  =  0. 
dx 

The  derivative  of  a  constant  times  a  function  is  equal  to  the 

constant  times  the  derivative  of  the  function  : 

[II]  d{c-u)^^    du 

dx  dx 

The  derivative  of  the  sum  of  two  functions  is  equal  to  the  sum 
of  their  derivatives : 

rjjj^  d(u  +  v)  _  du  .  dv  ^ 

dx  dx     dx 

The  derivative  of  a  power,  x^,  loith  respect  to  x  is  na;""^* 

[IV]  ^  =  nx"^-^. 

dx 

[We  shall  prove  this  at  once  in  the  case  when  n  is  a  positive  integer ;  later 
we  shall  prove  that  it  is  true  also  for  negative  and  fractional  values  of  ?i.] 

Each  of  these  rules  was  illustrated  in  Chapter  II,  §  17.  To 
prove  them  we  use  the  rule  of  §  17. 

Proof  of  [I].  If  ?/  =  c,  a  change  in  x  produces  no  change  in  y ; 
hence  A?/  =  0.  Therefore  dy/dx  =  lim  Ay/ Ax  =  lim  0  =  0  as  Ax 
approaches  zero.  Geometrically,  the  slope  of  the  curve  y  =  c 
(a  horizontal  straight  line)  is  everywhere  zero. 


30  ALGEBRAIC  FUNCTIONS  [III,  §  19 

Proof  of  [II].  liy  =  c  -u  where  m  is  a  function  of  x,  a  change 
Art'  in  x  produces  a  change  Ati  in  u  and  a  change  Ay  in  y; 
following  the  rule  of  §  17  we  find  : 

(A)  y  +  Ay  =  C'(u-\-  Au). 

(B)  A?/  =  c  •  Au. 

(C)  Ay/Ax  =  c  •  (Au/Ax). 

(D)  dy/clx  =  lim  [c  •  (Au/Ax)!  =  c  •  Urn  Au/Ax  =  c(du/dx). 

Ar=0  Ai=0 

Thus  d(7  x^)/dx  =  7  •  d{x')/dx  =  7  ■2x  =  Ux.    (See  §  §  4, 17. ) 

Proof  of  [III]-  If  y=u-\-v,  where  ?«  and  v  are  functions  of  x, 
a  change  A;c  in  x  produces  changes  Ay,  Au,  Av  in  y,  u,  v,  respec- 
tively, hence 

{A)         y  +  Ay={u  +  A^i)  +  {o  +  Av); 
(B)  Ay  =  Au  +  Av ; 

(O)         Ay /Ax  =  An /Ax  -\-  Ay/Ar ; 
(D)  dy/dx  =  lim  (Au/Ax)  +  lim  (Av/Aa;)  =  dxi/dx  +  dv/dx, 

Ax=0  Ai=0 

Thus 

d(a^-12a;4-7)^d(x^)      d(12x)   ^  d(7)^^^,      ^^  ^  ^^ 
dx  dx  dx  dx 

by  applying  the  preceding  rules  and  noticing  that  dx^/dx 
=  3  x^.  [See  Ex.  1  of  Exercises  VI  and  compare  Example  3, 
p.  10,  and  Example  2,  p.  25]. 

Proof  of  [IV].     If  y  =  x",  we  jjroceed  as  in  Example  5,  p.  26: 

(A)  y  +  Ay  —  (x  +  Ax)"  =  x" -^  nx'^-'^Ax -\-  (terms  which  have  a^ 

common  factor  Ax  ). 

(B)  Ay  =  nx^'^Ax  -f  (terms  with  a  common  factor  Ax  ). 

(C)  Ay / Ax  =  nx''-'^  +  (terms  which  have  a  factor  Ax). 

(D)  dy/dx  =  lim  (Ay /Ax)  =  nx"-^. 

Ax  =  0 

This  proof  holds  good  only  for  positive  integral  values  of  n. 
For  negative  and  fractional  values  of  n,  see  §§  20,  23. 


Ill,  §  19]  POLYXOMIALS  31 

Example  1.    (l{x^)/dx  =  9  x?.    (This  would  be  serious  without  the  rule. ) 
Example  2.         dx/dx  =  1  .  a;"^  =  1,  since  a;''  =  1. 

This  is  also  evident  directly:  dx/tte  =  lim  Ax/Ax  =  1.  Notice  how- 
ever  that  no  new  rule  is  necessary. 

Example  3.    —  (x*  -  7  x-  +  3  x  -  5)  =  4  x=^  -  14  a;  +  3, 
dx 

Example  4.    —  (^x"»  +  -B.':"  +  C)  =  mAz"'-'^  +  nBx''-\ 
dx 

EXERCISES  VII.  —  DIFFERENTIATION  OF  POLYNOMIALS 

Calculate  the  derivative  of  each  of  the  following  expressions  with  re- 
spect to  the  independent  variable  it  contains  (x  or  r  or  s  or  «  or  ?/  or  m). 
In  this  list,  the  first  letters  of  the  alphabet,  down  to  n,  inclusive,  represent 
constants. 

1.  (a)  2/ =  5x3.  (d)  y  =  5(x3  +  l).  (gr)  ?/ =-10xW  +  10. 
(6)  y  =  x*/4.              (e)  y  =  (x*  -  2)/4.  (h)  y  =  8  x^  +  6  x*. 

(c)  2/  =  5  .<-3  +  1.       (/)  2/  r=  -  10  xio.  (t)  2/  =  7  x6  -  6  x^  +  5. 

2.  (rt)  2/  =  «->-^-  (c)  2/  =  («  +  'j)  ■^•^.  (e)  1/  =  ^^-^  -  kx^  +  I- 
(6)  2/  =  -  c-^x9.          (d)  2/  =  (rt-  -  h'^)  xK  (/)  2/  =  ^  +  Bx  +  Cx2. 

3.  (a)  s  =  i-«-2.  (ft)  s  =  (2(3^ +  ^2).  (c)  s  =  c  (a<3  +  6«4), 

4.  (a)  g=s(s2-n.  (c)  g  =  (1  -  «3)  (2  +  gS). 
(6)  g  =  s^{a  -bs  +  cs"^).               (d)  q  =  as(b  +  cs)  +  d. 

5.  {a)  z  =  (y  +  a){y-b).  (c)  z  =  {y''-> +  2)(y^o  _  s). 
(6)  z  =  ay\y-  +  by^).                    (d)  z  =  (3  y'-  +  2)2. 

6.  (a)  v={hti*-ku^  +  l)u\  (b)  v  =  a  (rfi  +  u  +  l){ic^  -  u  +  I). 

7.  (a)  y  =  kx'*  +  /.<;"•.  (ft)  i/  =  a^e'-"  -  &•»""• 

8.  (a)  2/  =  a;2»+'"  +  x''+2'"  +  i.  (ft)  ?/  =  a  +  ftx"'. 

9.  Determine  the  slope  of  the  curve  y  =  x2  —  2  x  at  the  origin. 
Where  is  the  slope  2  ?  Where  is  the  tangent  horizontal  ?  Draw  the 
graph. 

10.  Locate  the  vertex  of  the  parabola  y  =  x"^  +  8  x  +  19  by  finding 
the  point  at  which  the  tangent  is  horizontal. 

11.  Proceed  for  each  of  the  following  curves  as  in  Ex.  10  : 

(a)  2/  =  x2  -  2  X  +  2.       (6)  J,  =  _  x2  +  2  X  -  10.       (c)  y  =  ax^  +  bx  +  c 


32  ALGEBRAIC   FUNCTIONS  [III,  §  19 

12.  Where  on  the  parabola  y  =  x^  is  the  slope  1  ?  Where  is  the  slope 
1  on  the  curve  y  =  x^?  Ou  ?/  =  x* '?  On  y  =  x"  ?  Where  is  the  slope  0 
on  each  of  these  curves  ? 

13.  What  is  the  slope  of  the  curve  ?/  =  2  x^  —  3  x^  +  4  at  x  =  0,  ±2, 
±4?  Where  is  the  slope  9/2?  —3/2?  Where  is  the  tangent  hori- 
zontal ;  are  these  points  highest  or  lowest  points,  or  neither  ?  Dravf  the 
graph. 

14.  What  is  the  slope  of  the  curve  y  =  xV4  —  2  x^  +  4  x^  at  x  =  0,  1, 

—  1,  —  2  ?    Where  is  the  tangent  horizontal  ;  are  these  points  maxima  or 
minima  ?     Where  is  the  slope  equal  to  eight  times  the  value  of  x. 

15.  Show  that  the  function  x^  +  Sx^  +  Sx  +  l  always  increases  with 
X.  Where  is  the  tangent  horizontal  ?  Show  that  there  is  no  maximum 
or  minimum  at  this  point. 

16.  Locate  the  maxima  and  minima  (if  any  exist)  on  each  of  the  fol- 
lowing curves  and  draw  their  graphs  accurately  : 

(a)  y  =  x^-27x+  15.  (d)  y  =  4x^-  11  x^  -  70x  +  20. 

(6)  y  =  2  x3  -  9  x2  +  12  X  -  10.  (e)    y  =  3  x*  -  4  x^  +  5. 

(c)    y  =  x3  -  9x2  -I-  27  X  -  15.  (y)  y  =  3x^-  80x3  +  iqoo. 

17.  At  what  angle  does  the  line  y  =  2x  meet  the  parabola  y  =  x^  + 
4x  +  l? 

18.  Find  the  angle  between  the  curve  y  =  x^  and  the  straight  line 
2/  =  9  X  at  each  of  their  points  of  intersection. 

19.  At  what  angles  does  the  curve  y  ={x  —  l){x  —  2){x  —  3)  cut  the 
X-axis  ? 

20.  If  a  sphere  expands  —  as  when  a  rubber  balloon  is  distended,  or 
when  an  orange  is  growing  —  the  volume  and  the  radius  both  increase. 
Find  the  rate  of  growth  of  the  volume  with  respect  to  the  radius. 

21.  In  an  expanding  sphere,  find  the  rate  of  growth  of  the  surface 
with  respect  to  the  radius. 

22.  Find  the  rate  of  change  of  the  total  surface  of  a  right  circular 
cylinder  with  respect  to  the  radius,  the  altitude  being  fixed  ;  with  respect 
to  the  altitude  when  the  radius  is  fixed. 

Do  the  same  for  a  right  circular  cone. 

20.  Differentiation  of  Rational  Functions.  In  order  to  dif- 
ferentiate all  rational  functions,  we  need  only  one  more  rule, 

—  that  for  differentiating  a  fraction. 


Ill,  §  20]  RATIONAL  FUNCTIONS  33 

Tlie  derivative  of  a  quotient  N/D  of  two  functions  N  and  D 
is  equal  to  the  denominator  times  the  derivative  of  the  numerator 
minus  the  numerator  times  the  derivative  of  the  denominator,  all 
divided  by  the  square  oj  the  denominator : 

'-*-'  dx     ~  D' 

To  prove  this  rule,  let  y  =  iVyZ),  where  N  and  D  are  func- 
tions of  x;  then  a  change  Ax  in  x  produces  changes  A?/,  AiV, 
AD  in  y,  N,  and  D,  respectively ;  hence,  by  the  rule  of  §  17 : 

JSr+AN 


(A)  y  +  Ay 


D  +  AD 


/„N  A        N+AN     N     D-AN-NAD 


{G) 


D  +  AD      D  D{D  +  AD) 

Ay  Ax  Ax 

Ai~    D{D  +  AD)  ' 


j)dJ[_^dD 

(D)  'll=lim^  =  -J^—^. 

dx       Ax^y)  Ax  D  - 


(3  X  -  7)  —  (x2  +  3)  -  (.v:2  +  3)  —  (3  .r  _  7) 

E^,nplel.    ilf±^W- "^        ,'      -,.       "• 

dx\3x-7j  (3a:-  0" 

^  (3x-7)-2x-(x^+S)-S  ^  3  3-2-14X-9 
(3a;-7)2  (3a;-7)2    * 

TT  7    o         ^  /  1  \  dx  dx        0  -  2  X  2 

Example  2.      —(  —  )=: — — =  - — tL±=  —  ±. 

dx\xy  (x2)2  X*  a;3 

(Compare  Example  3,  §  17,  p.  25.) 

Example  3.       ^  (a^*)  =^f  lU  ^ziiL^  =  ^=  -kx-'-\ 
dx         '     dxXx")  x2*  a-*+i 

Note  that  formula  IV  holds  also  when  n  is  a  negative 
integer,  for  if  n  =  —  A;,  formula  IV  gives  the  result  we  have 
just  proved. 


34  •  ALGEBRAIC  FUNCTIONS  [III,  §  20 

EXERCISES   VIII.     DIFFERENTIATION   OF   RATIONAL  FUNCTIONS 

Calculate  the  derivative  of  each  of  the  following  : 

1.    (a)  2/  =  ^-  (e)   y^'^-^^. 


x  +  4. 


w.=^  +  !- 

X—  1 

<")-!f^- 

(d)  2/=-^- 

1  +  x 

2. 

(-)y=~ 

'''y=^^- 

(c),  =  x-(  =  l) 

(d)  y^^x'K 

(e)   y  =  ^x-\ 

3. 

ia)v--\-\. 

u^  —  1 

(^)-'-^- 

^^^-^.■ 

id)  v=-l-^. 

u  -- 

u 

4. 

(»)«  =  <,+5_£. 

<')-^ri7- 

(c)    s  =  ht^-kt-^. 

id)p-r^  '•:+; 

if)y-^'^' 


(9)  y 


a;2-  1 

X2  +  x-3 


x2  -  2  X  +  6 

(fif)  y  =  2x-3-3x-2. 

(A)  y  =  4x-6  +  4- 

X* 

(»■)    ?/  =  8.r-w  +  ^-15. 


(j)  y 

=  ax-"*  +  bx-". 

(e)    tJ 

u-  —  tl  +  1 

(/)s 

]                             1 

t^  +  2t-S      f'  +  t  +  6 

{g)  s 

2t+  1             5 

«3  -  1          fi^t+l 

(h)   s 

=  a+     ^    . 

'^\ 

(e)    s 

^t-^{t^-  2<5  +  l). 

{/)<! 

=   (S-3+4)(s-3_5). 

as-* 

{g)  1 
(h)  p 


bs^  +  c 

j-S  _  J.2 


3  +  6x-5x2  /,>,   ■,^g^/4  +  2g3_ 

X  ■  ^  ^  ^  2  ff^/7 

3  xV2  ^  ^2  +  2 . 


5.    («)2/  = (c)p=        ^^^^^ 


Ill,  §  21]  PRODUCT  35 

6.  Draw  the  following  curves  ;  obtain  the  equation  of  the  tangent  at 
the  point  indicated,  and  also  at  any  point  (xq,  yo)  ;  determine  the  hori- 
zontal tangents  if  any  exist,  and  show  whether  these  points  correspond  to 
maxima  or  minima  or  neither. 

1  +  X  X 

7.  Compare  the  slopes  of  the  curves  y  =  x,  y  =  x-^  at  the  points  at 
which  they  intersect.     What  is  the  angle  between  them  ? 

8.  Compare  the  slopes  of  the  family  of  curves  y  =  x",  where  n  =  0, 
+  1,  +  2,  etc.,  —1,-2,  etc.,  at  the  common  point  (1,  1).  What  is  the 
angle  between  ?/  =  x-  and  y  —  x~^  ?     See  Tables,  III,  A. 

21.  Derivative  of  a  Product.  —  The  following  rule  is  often 
useful  in  simplifying  differentiations  : 

TJie  derivative  of  the  product  of  two  functions  is  equal  to  the 
first  factor  times  the  derivative  of  the  second  plus  the  second 
factor  times  the  derivative  of  the  first : 

dx  dx  dx 

li  y  =  u  •  V  where  u  and  v  are  functions  of  x,  a  change  Aa:  in 
X  produces  changes  Ay,  Aw,  Ay  in  y,  u,  and  v,  respectively: 

{A)       y  +  Ay  =  {u  +  A  «)  (v  +  Ay) ; 

(B)  Ay  =(u  +  A.u)(y  +  Av)— m  -  v  =  uAv  -}-v  A/t  +  Au  Au ; 

(  C)       A.y/ A.'c  =  u  (Av/Ax)  +  v(A?//Aa;)  +  A?<  —  ; 
(D)        dy/dx  =  lim  (Ay /Ax)  =  n  (dv/dx)  +  v(du/dx). 

Aa:=0 

Example  1.    To  find  the  derivative  of  y  =  (x^  +  3)  (.r^  +  4). 

Method  1.     We  may  perform  the  indicated  multiplication  and  write: 

^  =  A  [ (x2  +  3)(x3  +  4)  ]  =  i^  [x5  +  3  x8  4-  4  x2  +  ] 2]  =  5  X*  +  9  x2  +  8  a;. 
ax     ax  dx 


36  ALGEBRAIC  FUNCTIONS  [III,  §  21 

Method  2.     Using  the  new  rule,  we  write  : 

^  =(x2  +  S)4-(x^  +  4)  +  (x3  +  4)  I-  (x2  +  3) 
dx  dx  dx 

=  [y?  +  3)3  x2  +  (x3  +  4)2  x=5  X*  +  9  x2  +  8  x. 
In  other  examples  which  we  shall  soon  meet,  the  saving  in  labor  due 
to  the  new  rule  is  even  greater  than  in  this  example. 

22.  The  Derivative  of  a  Function  of  a  Function.  Another 
convenient  rule  is  the  following : 

The  derivative  of  a  function  of  a  variable  ii,  ivhich  itself  is  a 
function  of  another  variable  x,  is  found  by  multij)lying  the  deriva- 
tive of  the  original  function  loith  respect  to  u  by  the  derivative  of 
?*  loith  respect  to  x. 

[VH]  dy  ^dy    du 

dx      du    doc 

If  y  is  a  function  of  u,  and  m  is  a  function  of  x,  a  change  Acb 
in  X  produces  a  change  A?/-  in  u;  that  in  turn  produces  a 
change  A?/  in  ?/ ;  hence : 

Aj/_  A// ^  Au^ 
Ax      Aii    Aa; 

Taking  limits  on  both  sides,  we  find : 

dy  _dy  du^ 
dx     du.  dx 

This  is  really  the  same  as  the  rule  used  in  §  8,  p.  14 ;  for,  if 
we  divide  both  sides  by  du/dx,  we  find 

[Vila]  rly^rJM^^* 

du      dx      dx 

which  is  the  same  as  the  rule  of   §  8,  except  that  different 
letters  are  used. 

Example  1.    To  find  the  derivative  of  j/  =(x2  +  2)^ 

Method  1.     We  may  expand  the  cube  and  write  : 

^  =  A  [  (x2  +  2)31  =  -^  (x6  +  6  X*  +  12  x2  +  8)  =  6  x5  +  24  x3  +  24  x. 
dx     dx  dx 


Ill,  §  22]  FUNCTION   OF  A  FUNCTION  37 

Method  2.     Using  tlie  new  rule,   we    may   simplify   this  work :    let 
w  =  x2  4-  2,  then  y  =  (a;^  +  2)8  =  u^ ;  rule  [VI]  gives 

dx     du    dx        du  dx 

=  3(x2  +  2)2.  (2a;)  =  3(a:«  +  4x2  +  4)  •  (2  x)  =  6x5  +  24  x8 +  24  x. 
Example  2.     liy  =  f^  +  2  and  x  =  3  <  +  4,  to  find  dy/dx. 
Method  1.     We  may  solve  the  equation  x  =  3  f  +  4  for  <  and  substitute 
this  value  of  t  in  the  first  equation : 

^      \,    3     ;    ^  9      9^9 

Method  2.     Using  the  new  rule  (with  letters  as  used  in  §  8,  p.  14) 
we  write  : 

dy_di^dx^  d(t^  +  2)  ^  d{S  t  +  4)  ^  2t  ^S  =  -t 
dx     dt   '  dt  dt        '         dt  ■  '       3  ■ 


EXERCISES  IX.  — SHORT  METHODS.  RATIONAL  FUNCTIONS 

Calculate  the  derivative  of  each  of  the  following : 

1.  (a)  2/=3x(x2+ 1).  (d)  y  =  (2x+l)(l-x  +  x2). 
(^)  ?/ =  x3(x2  +  3).                          (e)    2/  =  (x2-4)(l +x3). 

(c)   ?/  =  (3x  +  2)(2x-3).  (/)  2/  =  (x3  +  3x-2)(x2-2x). 

2.  (a)  y=  (x2  +  1)2.        -    (c)  y  =  (1  -  x2)2.  (g)  y  =  {a  +6x)». 
(6)  2/  =  (x2  -  1)3.            (d)  y  =  (1  -  x2)3.  (/)  y={a  +  bx)K 

3.  (ffl)  y  =  (1  +  2  X  -.  3  x2)2.  {d)  2/  =  («  +  6x  +  cx2)3. 
(i>)  y  =  (x-i  + 3x  +  7)3.  (e)  s  =  (3«2  + 2«-4)*. 
(c)   s  =  («3  _  f  _  4)2.                  (/)  2/  =  (a  +  6x  +  cx2)5. 

4.  (a)  y= ^ (d)  y  =  (2 +3x2)-2r= ?— t-:1- 

*"   '   ^      (l+2x- 3x2)2  ^   ^  ^       ^    ^       ^     L     (2  +  3x2)2j 

(b)    s= (e)  »  =  (a  +  6x)-8. 


(x2  +  2)3  (a  _  6s  -  cs2)3 

5.    (a)  2/=  (1  -5x2)(3-4xS)(l-x).       (c)  2/ =  (a;2  +  2)3(3x  -  5)^ 
(6)  2/  =  a:(x2  +  3)(x3  +  4).  (d)  s  =  (<3  _  2)2(2  «- l)". 


38  ALGEBRAIC  FUNCTIONS  [III,  §  22 

6.    Determine  dyjUx  in  each  of  the  following  pairs  of  equations  : 

^^    |M  =  3a;-4.  ^   ^     1  m  =  x2  _  1/2. 

r  6;s-4^  [2^-4 


2=2-4x. 


4x8 


4x 


7.  Draw  each  of  the  curves  represented  by  the  following  pairs  of 
parameter  equations  and  determine  dy/dx  : 

,  .     fx  =  «2,  ...      fx  =  2f  +  3«2, 

(«)    \y=3t  +  2.  (^)     [y  =  2t+i. 

What  is  the  slope  in  each  case  when  t  =  1?  Show  this  in  your  graphs. 
Find  the  value  of  the  slope  in  each  case  at  a  point  where  the  parameter 
has  the  value  2. 

8.  Draw  the  graph  of  the  function  y  =  (2x  -  l)2(3x  +  4)2.  De- 
termine its  horizontal  tangents. 

9.  Proceed  as  in  Ex. '8,  for  the  function?/ =(2  x  -  1)2-- (3  x  +  4)2. 

10.  Show  that  if  y  =(x  -  1)2(2  x  +  3)2,  the  derivative  dy/dx  has  a 
factor  (x  —  1)  and  a  factor  (2  x  +  3) ;  hence  show  that  the  given  equation 
represents  a  curve  tangent  to  the  x-axis  at  x  =  1  and  at  x  =  —  3/2. 

11.  Show  that  if  2/  =  (x  -  2)3(x3  +  4x  —  7),  the  derivative  dy/dx  has 
a  factor  (x  —  2)2.  Show  that  the  given  curve  is  tangent  to  the  x-axis  at 
X  =  2,  but  has  no  minimum  or  maximum  there. 

12.  Apply  the  same  reasoning  which  was  used  in  Ex.  11  to  the  equa- 
tion y  =  {x  —  aY{x  —  by. 

13.  Show  that  the  curve  j/  =  x^  +  «x2  +  6x  +  c  is  tangent  to  the  x-axis 
at  x  =  k  if  (x  —  A:)2  is  a  factor  of  the  right-hand  side. 

14.  Show  that  y  =  P  (x)  where  P  (x)  is  any  polynomial,  is  tangent  to 
the  X-axis  at  x  =  A;  if  (x  —  k)^  is  a  factor  of  P(x). 

23.  Differentiation  of  Irrational  Functions.  In  order  to 
differentiate  irrational  expressions,  we  proceed  to  prove  that 
the  formula  for  the  derivative  of  a  power  (Rule  [IVJ)  holds 
true  for  all  fractional  powers : 

nv  n  <*a^"  Hi  .   P 


Ill,  §23]  IRRATIONAL  FUNCTIONS  39 

First  iiroof.  Let  y  =  a;p/«,  where  x>  and  q  are  positive  integers ; 
then,  raising  both  sides  to  the  power  q, 

(1)  y^^x^. 

If  {x,  ?/)  and  {x  +  A.r,  y  +  A?/)  are  pairs  of  related  vahies  of 
X  and  y,  each  pair  must  satisfy  equation  (1) ;  hence  (1)  holds 
for  X  and  y,  and  also 

(2)  (2/  +  Ay)«  =  (.K  +  AaO'', 


(3)     y^  +  q  •  y'^^y  +  (several  terms)  a/ 

=  a;"  +p.i"'~^Ax'  -I-  (several  terms)  A.r  . 

Subtracting  (1)  from  (3),  and  dividing  both  sides  of  the  result- 
ing equation  by  Ax : 

[g?/«~^  +  (several  terms)  A?/]  -^  =2)x''^^  +  (several  terms)  Ax, 
Ax 

Ay  _  j)xp~^  +  (several  terms)  Ax 
Ax      Qi/9~^  +  (several  terms) Ay' 
whence 

^=  lim  (^\  =  P^^     P^"-'     =Pxv/a-y, 
dx      A===o\^Axy       gy«-^      qixp^'^y-'^      q 

This  is  the  same  as  formula  [IV]  with  ?«  =  -;  hence  [IV] 
holds  for  any  positive  fractional  exponent. 

That  [IV]  also  holds  for  negative  fractional  exponents  is  now 
proved  by  means  of  Ex.  3,  p.  33;  hence  [IV]  holds  for  any 
positive  or  negative  fractional  exponent. 

Second  proof.  Another  proof  will  seem  simpler  to  some 
students :  if  we  set 

(1)  X  =  t",  then  y  =  t^, 

which  together  are  equivalent  io  y  =  x^''',  and  apply  formula 
[Vila]  with  suitable  changes  of  letters,  we  find: 

dy^dy^dx^        ,  ^       ,  ^  22  f,-, 
dx      dt      dt  Q 


40  ALGEBRAIC  FUNCTIONS  [III,  §  23 

but  since  t  =  x^^^,  substitution  for  t  gives 

dx      q  q 

This  proves  [IV]  for  positive  fractional  values  of  n ;  the  proof 
for  negative  fractional  exponents  is  as  given  in  Ex.  3,  p.  33. 

The  rule  also  holds  when  n  is  incommensurable ;  for  example, 
given  y=x'^^,  it  is  true  that  dy/dx  =  ^2x'^^~'^;  we  shall  post- 
pone the  proof  of  this  until  §  84,  p.  147. 

24.  Collection  of  Formulas.  Any  formula  may  be  combined 
with  [VIIJ,  for  in  any  example,  any  convenient  part  may  be 
denoted  by  a  new  letter,  as  in  §  22.  For  example,  Rule  [IV] 
may  be  written 

^"  =  ^  .  ^,  by  [VII],        =nn'^-'  .  ^,  by  [IV]. 
dx       du     dx'    -^  •-       j»  dx     ^  ^      -^ 

The  formulas  we  have  proved  are  collected  here  for  easy 
reference : 

[11] 


dx  dec 

[HI]  d(u  +  v)^^_^^     Holds  for  subtraction  also. 

doc  doc      doc 

[IV]  ^  =  nt."-if^. 

doc  doc 

a(¥.]     D^-N^  d'-     -cf 

rv'T  V-D/  doc  doc       o  •    1  ^*  "^ 

ryi]  d{u   v)^^dv,^dM^ 

doc  doc        doc 

Special  case:    —  =  1.     (y  =  x.)  (I  19) 

dx 


Ill,  §  25]  IRRATIONAL  FUNCTIONS  41 

These  formulas  enable  us  to  differentiate  any  simple  alge. 
braic  function. 

25.  Illustrative  Examples  of  Irrational  Functions.  In  this 
article  the  preceding  formulas  are  applied  to  examples. 

Examplel.    ^  =  ^'=  ia:V2-i  =  L-^/z  =^  .    (See  Ex.  4,  p.  25.) 
dx         dx        2  2  2v'x 

Example  2.     Given   y  =  VS  x-  +  4,  to  find  dy/dx. 

Method  1.     Set  M  =  3  a;2  +  4,  then  y=^u;  by  Rule  [VII], 
dy_dy^du  _     1        g     _        6  a;        _  3  x  v^3  xi^  +  4 
dx     du    dx     2\/m  2\/3x2  +  4  3x^  +  4 

Method  2.  Square  both  sides,  and  take  the  derivative  of  each  side  of 
the  resulting  equation  with  respect  to  x  : 

dCy^)^d(3x^jM)^6x. 
dx  dx 

But  by  Rule  [IV], 

d(y^^d(y^  .  %  =  2y^; 
dx  dy        dx  dx' 

hence, 

2y^  =  6x,  or^^3x^_3x       ^SxVS^^+l 
dx  '        dx       y       V3  x2  +  4  3  x2  -1-  4 

This  method,  vs-hich  is  excellent  when  it  can  be  applied,  can  be  used  to 
give  a  third  proof  of  the  Rule  [IV]  for  fractional  powers.  The  next 
example  is  one  in  which  this  method  cannot  be  applied  directly. 

Example  3.     Given  y  =  a:^  _  2  V3  x^  +  4,  to  find  dy/dx. 


dy^d(x^)      2'^   V3^n~i=3x'      6xV3x-^  +  4 
dx        dx  dx  3x2  +  4 


Example  4.     Given  y  =  (x^  -  2)  VS  x^  +  4,  to  find  dy/dx. 

^  =  V3x-i  +  4-!^(x8i-2)  +  (x3-2)A(V3x2  +  4)  [by  Rule  VI] 

dx  dx  dx       ^ 


V3  X-  +  4  .  3  x2  +  (x8  -  2)  3  a:  V3  x2  4-  4  .j^^  Example 

3x*+  4 


=  V3^H4r3x2  +  (x3-2).-^^1  =  V3^M^  .  12£i±12x^z:6^, 
L        ^  ^    3x2-|-4j  3x2  +  4 

Example  5.     Given  ?/  =  ^ a;  +  I-Vx^  ^^  ^^^^  di//dx. 
.Vx  +  1  +  Vx 


42  ALGEBRAIC   FUNCTIONS  [HI,  §  25 

First  reduce  y  to  its  simplest  form  : 

Vx+l-Vx     Vx+l— Vx      2a;+l-2Vx^ 


'"viTl 

"+Vx 

ViC  +  1- Vx 

Then 

f  C2X  +  1)- 
dx 

-2- 

where  m  = 

X2  +  X 

;  hence 

dy. 

dx 

=  2       2     1 

2  vT 

dx 

(x+l)-x 


2x  +  l-2Vx2+x. 


dx  du    dx 


(2x  +  l). 


V  X^  +  X 

This  example  may  be  done  also  by  first  applying  the  rule  for  the  deriva- 
tive of  a  fraction  [Rule  V]  ;  but  the  work  is  usually  simpler,  as  in  this 
example,  if  the  given  expression  is  first  simplified. 


EXERCISES   X.— ALGEBRAIC   FUNCTIONS 

Calculate  the  derivatives  of 


1.    (a)  y  =  x4/3.                   {d)  y  =  Vx^. 
{h)  s  =  10^5/2,                (e)    s  =  2V¥\ 
(c)   y  =  x^'K                   if)  v=i^u\ 

(/i)  2/  =  6x-2/3. 

2.    {a)y  =  x-    '_      '''^               (b)v-. 

Vx«      Vx5 

(C)    S  =  <3(2«2/3  +  3^-2/3).              (^a)    y 

_  6  _    2 

=  2v/x(xi/3  +  a;6/3). 

U             23^r3J 

3.    2/  =  f  f  X  \/x5  -  if  x2  Vx^  +  i^  x3  y/x. 

4.    (a)  ?/=  V2  +  3X. 

(Sr)  2/  =  ^1  +  xK 

(6)s  =  V3f-4.  (A)  2/=  V2x2  +  4x. 

(c)  i;  =  M\/2  +  3  m.  (0    ?/  =  x^-v/Sx  — 4. 


(d)s=v/«2-l.  (j)  2/  =  (5  +  3x)V6x-4. 

(e)   s  =  V<^  -  3 1.  (k)  V  =  Vl  -  X  +  xK 

(/)  s  =  -A+-3j-  (Z)  s  =.  V6  t  -  5. 
Ve  (  -  5  5  +  3 « 


5.    (a)  2/  =  V  1  +  Vx.         (ft)  s  =  \y-^-  (<=)  " 


Ill,  §  25]  ALGEBRAIC   FUNCTIONS  43 


(a)  y^{9-6x  +  5x2)V{l  +  x*y.  (b)  s  =  (I  +  r-)  y/T^~ti, 

(b)  r  =  f  A  +  1\  V(3  -  5w2)6. 

— ;z^i~-       (First  rationalize  the  denominator.) 
Vl+x+Vl-x 


(,^^^V1  +  X^  +  X.  (c),  =  ^^«^^. 

Vl  +  x2  —  a;  Va^  +  a;^ 

V(20  — 3x»)"-' 

10.    Draw  the  graphs  of  the  equations  below,  and  determine  the  tan- 
gent at  the  point  mentioned  in  each  case. 


(ffl)  2/ =  VT^,  (X  =  I).        {d)  y=  V(r+x)(2  +  3x),(x  =  2). 
(6)  y=  Vl+x2,  (x  =  |).        (0  2/  =  xVrT^,  (x  =  l). 
(c)  2/  =  Vx,  (a:  =  2).  (/)  ?/  =  xV^  -  xV3,  (x  =  1). 

11.  Find  the  angle  between  the  curves  y  =  x^/^  and  y  =  x^  at  each  of 
their  points  of  intersection. 

12.  Find  the  angle  between  the  curves  y  =  .r^/s  and  y  =  x^/^  at  (1,  1). 

13.  Find  the  angle  between  the  curves  y  =  xp/«  and  y  =  x«/p  at  (1,  1). 

14.  In  compressing  air,  if  no  heat  escapes,  the  pressure  and  volume 
of  the  gas  are  connected  by  the  relation  pv^-*^^  =  const.  Find  the  rate  of 
change  of  the  pressure  with  respect  to  the  volume,  dp/dv. 

15.  In  compressing  air,  if  the  temperature  of  the  air  is  constant,  the 
pressure  and  the  volume  are  connected  by  the  relation  jiv  =  const.  Find 
dp/dv,  and  compare  this  result  with  that  of  Ex.  14. 

16.  Find  dy/dx  for  y  =  x-  ;  for  y  =  x--^  ;  for  y  =  x^-s  ;  for  y  =  x^-*  ;  for 
y  =  x^.  Show  that  the  value  of  dy/dx  increases  steadily  in  each  case  as 
X  increases,  and  that  the  magnitudes  of  the  derivatives  are  in  the  order  of 
the  exponents  at  x  =  1  and  for  all  larger  values  of  x. 

17.  Draw  a  graph  to  show  the  values  of  the  derivatives  for  each  of  the 
curves  of  Ex.  16  ;  find  graphically  the  values  of  x  for  which  the  derivative 
of  each  of  them  is  the  same  as  that  of  y  —  x-. 


44  ALGEBRAIC   FUNCTIONS  [III,  §  26 

PART  II.     EQUATIONS   NOT  IN   EXPLICIT   FORM 
DIFFERENTIALS 

26.  Solution  of  Equations.  An  equation  in  two  variables  x 
and  y  is  often  given  in  unsolved  form ;  i.e'  neither  variable  is 
expressed  directly  in  terms  of  the  other.     Thus  the  equation 

(1)  a^-^y'  =  l 

represents  a  definite  relation  between  x  and  y ;  graphically,  it 
represents  a  circle  of  unit  radius  about  the  origin. 

Such  an  equation  often  can  be  solved  for  one  variable  in 
terms  of  the  other ;  thus  (1)  gives 

(2)  y  =  Vl  —  X-,  or  y  =  —  y/1  —  x\ 

The  first  solution  represents  the  upper  half  of  the  circle,  the 
second  the  lower  half.  From  this  solution,  we  can  find  dy/dx 
as  in  §  25 : 

(3)  dy^_-^x_^  ^^  dy^      +x     ^ 
dx      Vl— ar^  ^^'      Vl-ar' 

where  the  first  holds  true  on  the  upper  half,  the  second  on 
the  lower  half,  of  the  circle. 

By  Rule  [VII]  such  a  derivative  may  be  found  directly 
without  solving  the  equation.     From  (1) 

dx  dx 

but  ±(x^  +  f)=^(^  +  'Mjn=2x  +  '^(fl  .^,      by  VII; 

dx  ^       dx         dx  dy       dx        '' 

hence 

(4)  2a.'  +  22/g  =  0, 

or 

(5)  ^  =  _^. 
^  ^  dx         y 

This  result  agrees  with  (3),  since  y  =  ±  Vl  —  x^. 

This  method  is  the  same  as  that  used  in  the  first  proof  of 
[IVa]  in  §  23,  p.  39,  and  also  in  the  second  solution  of  Ex.  2, 


Ill,  §  27]  IMPLICIT   FUNCTIONS  45 

p.  41.     It  may  be  used  whenever  the  given  equation  really  has 
any  solution,  without  actually  getting  that  solution. 

Such  a  formula  as  (4)  is  much  more  convenient  than  (3), 
since  it  is  more  compact,  and  is  stated  in  one  formula  instead 
of  in  two.  But  the  student  must  never  use  (5)  for  values  of  x 
and  y  without  substituting  those  values  in  (1)  to  make  sure 
that  the  point  (x,  y)  actually  lies  on  the  curve;  and  he  must 
never  use  (5)  when  (5)  does  not  give  a  definite  value  for 
dy/dx*  Thus  it  would  be  very  unwise  to  use  (4)  at  the  point 
x  =  l,  y  =  2,  for  that  point  does  not  lie  on  the  curve  (1) ;  it 
would  be  equally  unwise  to  try  to  substitute  x  =  l,  y  =  0,  since 
that  would  lead  to  a  division  by  zero,  which  is  impossible. 

27.  Explicit  and  Implicit  Functions.  If  one  variable  y  is 
expressed  directly  in  terms  of  another  variable  x,  we  say 
that  y  is  an  explicit  function  of  x. 

If,  as  in  §  26,  the  two  variables  are  related  to  each  other  by 
means  of  an  equation  which  is  not  solved  explicitly  for  y,  then 
y  is  called  an  implicit  function  of  x.  Thus,  (1)  in  §  26  gives  y 
as  an  implicit  function  of  x ;  but  either  part  of  (2)  gives  y  as 
an  explicit  function  of  x. 

If  an  equation  in  x  and  y  is  given,  so  that  y  is  an  implicit 
function  of  x,  we  may  either  solve  that  equation  for  y,  as  in 
the  first  part  of  §  26,  and  then  differentiate  as  we  have  done  up 
to  this  point ;  or  we  may  proceed  to  find  the  derivative  with- 
out solving,  by  means  of  Rule  [VII],  as  in  §  26.  The  latter 
method  is  especially  fortunate  when  the  given  equation  is 
difficult  to  solve. 

Definition.  If  the  original  equation  is  a  simple  polynomial 
in  X  and  y  equated  to  zero,  any  explicit  function  of  x  obtained 
by  solving  it  for  y  is  called  an  algebraic  function.     See  §  18. 

*  These  precautions,  which  are  quite  easy  to  remember,  are  really  suflScient 
to  avoid  all  errors  for  all  curves  mentioued  in  this  book,  at  least  provided  the 
equation  like  (4)  [not  (5)]  is  used  in  its  original  form,  before  any  cancellation 
has  been  performed. 


46  ALGEBRAIC   FUNCTIONS  [III,  §  27 

Example  1.    x'  +  j/^— 3xi/=0.    (Folium  of  Descartes  :   Tables,  111,  h.) 

This  equation  is  difficult  to  solve  directly  for  y.  Hence,  as  in  §  26,  we 
find  dy/dx  by  Rule  [VII]  ;  differentiating  both  sides  with  respect  to  x, 
we  find  : 

3x2 +  32/2  ^^-32/-3x^  =  0; 
dx  dx 

whence  dy  ^y_-^ 

dx      2/2  —  X 

At  the  point  (2/3,  4/3),  for  example,  dy/dx  =  4/5  ;  hence  the  equation 
of  the  tangent  at  (2/3,  4/3)  is  (2/-4/3)  =  (4/5)  (x-2/3)  or  4  x-5  »/+4=0. 
Verify  the  fact  that  the  point  (2/3,  4/3)  really  lies  on  the  curve.  Note 
that  this  formula  is  useless  at  the  point  (0,  0)  although  that  point  lies  on 
the  curve. 

28.  Inverse  Functions.  If  y  is  given  as  an  explicit  function 
of  X, 

(1)  y=fi^), 

and  if  this  equation  can  be  solved  for  x  in  terms  of  y, 

(2)  X  =  cl>(y), 

then  <})(y)  is  called  the  inverse  function  of  f(x).  If  this  solu- 
tion is  substituted  in  the  original  equation  (1),  that  equation 
must  be  satisfied: 

(3)  y=fWy)}' 

Thus,  it  y  =  x^;  we  find  x  =  y^'^\  substituting  ?/^^'  for  x  in 
the  original  equation  gives  y  =  {y^'^Y,  which  is  an  identity. 


Since 
it  follows  that 

^  =  1  _=.  ^, 

Ax           '  Ay 

[VII 6] 

^  =  1  -^  ^, 
doc          '  dy 

unless  dx/dy  —  0.*     This  rule  is  really  a  special  case  of  Rule 
[VII]  ;  for  if,  in  Rule  [VII],  y  =  x,  we  get 

*  The  precautions  to  be  observed  are  exactly  the  same  as  those  of  §  26. 


Ill,  §29]  IMPLICIT   FUNCTIONS  47 

dx     du  _  ^ 
du     dx 

which  agrees  with  [VII  &]  except  that  different  letters  are  used. 

Thus  if  n  =  x^,  x  =  ii^'^  ;  du/dx  =  3  x^  dx/du  =  1/(3  m^/s)  ;  then 
(du/dx)  •  (dx/du)  =  (3  x'^)  •  [1/(3  m^/^)]  =  1  since  x  =  i(V3. 

29.   Parameter  Forms.     If  both  x  and  y  are  given  as  explicit 

functions  of  a  third  variable  t : 

(1)  x=f{t),y  =  <t>(t), 

we  call  t  a  parameter,  and  the  equations  (1)  parameter  equa- 
tions. If  we  can  eliminate  t,  we  obtain  an  equation  connecting 
a;  and  y  directly  : 

(2)  F(x,  y)  =  0. 

From  (2)  we  might  find  dy/dx  as  in  §  26 ;  but  it  is  usually- 
easier  to  proceed  as  in  §  8,  p.  14,  and  §  22,  p.  36,  using  the 
formula  [Vila],  in  the  letters  x,  y,  t: 

rVTT    1  dy_dy^dx 

^        "^  dx~  dt    '   dt'. 

Thus  in  Example  2,  p.  37,  we  found  dy/dx  by  thi.s  formula  from  equa- 
tions like  (1)  ;  first,  by  eliminating  t ;  second,  by  using  [Vila]- 

EXERCISES  XL  — FUNCTIONS   NOT  IN   EXPLICIT  FORM 

In  each  of  these  exercises  the  student  should  take  some  point  on  the 
curve,  and  find  the  equation  of  the  tangent  there. 

1.  From  the  equation  xy  =  \  find  dij/dx  by  the  two  methods  of  §  26, 
first  solving  for  y,  then  without  solving  for  y.  Write  the  result  in  terms 
of  X  and  y  ;  and  also  in  terms  of  x  alone,  when  possible. 

2.  Find  dy/dx  in  the  following  examples  by  the  two  methods  of  §  20  : 

(a)  x-^y  =  10.  '    (/)  x2  -  2x2/  +  2x  -  3?/  +  4  =  0. 

(b)  x2  +  xi/  -  5  =  0.  (g)  x3  -  x^y  -4  =  0. 

(c)  x2  -  yi  =  1.  (h)  x3  -  2/2  =  0. 

(d)  xy  +  x  +  y  =  0.  (0  x^  +  y^  =  a^. 

(e)  4  x2  -  2/2  =  16.  0')   a:8  -  j/3  =  a\ 


48  ALGEBRAIC   FUNCTIONS  [III,  §  29 

3.  Find  dyjdx  in  the  following  examples  without  solving  for  y  ;  check 
the  answers  when  possible  by  the  other  method  of  §  26  : 

(a)  x2  +  3  xt/  +  2/2  =  2.  (c)  ay?  +  2  hxy  +  hy-  =  A;. 

(6)  JcV  _|.  2  x</  +  7  =  0.  (d)  y<  -  2  2/%  +  x-  =  0. 

(e)  ax2  +  2  6x!/  +  C2/2  +  2  (Zx  +  2  e^  +  /  =  0. 

(/)     V^  +   Vy  =   Va.  (^)    x3/2  +  2/3/2  =  (^3/2_ 

4.  Find  the  inverses  of  the  following  functions  by  solving  the  equations 
for  %  ;    then  find  dx/dy.     Verify  that  {dx/dy)  (dy/dx)  =  1  in  each  case. 

(a)    2/  =  2x  +  3.  ^^^   y=       ^i       . 

(6)   2/  =  5-x.  Vl+x 

(O    2/=""'  '   ' 


(c)    y=~ —  ^^  ex  +d 

0')  2/  =  x3. 

(A;)  y  =  Jx2  +  3. 

(0  2/ =  3x2 -5. 

(m)  2/  =  a:^  +  2. 

(0)  2/=(x-l)(x  +  2). 
5.   In  the  following  examples  find  dy/dx  without  solving  for  y : 

^  ^  32/-1 


(d) 

y 

=  6- 

2 

x' 

(0 

y 

_  2- 
2x 

-  X 

+  3" 

(/) 

y 
y 

=vr 

=  Va 

-X2. 

(S') 

^+X^ 

(6)    x  =  2/^-22/  +  4.  (?)    ^  =  2/-^+^^ 

(c)    X  =  2/^  +  5. 

Cd)   x  =  2/3-3y  +  7. 

(,)   :,  =  yl±ULlll. 

2/2  +  2 

6.  In  the  following  pairs  of  parameter  equations,  find  dy/dx  by  §  29  ; 
when  possible  eliminate  t  to  find  the  ordinary  equation,  and  show  that 
the  derivative  found  is  correct. 

^M2/  =  16f2.  '^  ^    l2/  =  3«  +  2.  ^M2/  =  2«3. 


Ill,  §  30]  RATES  49 

7.  In  each  of  the  problems  of  Ex.  6,  find  the  horizontal  speed  and 
the  vertical  speed  of  a  body  which  moves  as  stated  there,  x  and  y  repre- 
senting the  coordinates  of  the  body  at  the  time  t.  The  total  speed  along 
the  curve  is  the  square  root  of  the  sum  of  the  squares  of  these  two  ;  find 
this  total  speed  in  each  case. 

8.  In  a  circle  of  unit  radius  about  the  origin  dy/dx  =  —  x/y ;  this  is 
positive  when  x  and  y  have  different  signs,  negative  when  x  and  y  have 
the  same  sign.  Show  that  this  agrees  with  the  fact  that  the  circle  rises 
in  the  second  and  fourth  quadrants  and  falls  in  the  first  and  third  quad- 
rants as  X  increases. 

9.    Show  that  the  curve  xy  =  1  is  falling  at  all  its  points. 

10.  Show  that  the  curve  x^y  =  1  is  rising  in  the  second  quadrant  and 
falling  in  the  first  quadrant. 

11.  The  equation  x'/^  +  yi/2  =  i  is  the  equivalent  to  the  equation 
x^  —  2xy  +  y^  —  2x  —  2y  +  l=0,  if  the  radicals  ^1/2  and  ?/i/2  be  taken 
with  both  signs.  Show  that  the  values  of  dy/dx  calculated  from  the  two 
equations  agree.  By  methods  of  analytic  geometry,  it  is  easy  to  see  that 
the  curve  is  a  parabola  whose  axis  is  the  line  y  =  x,  with  its  vertex  at 
(1/4,  1/4). 

12.  The  curve  of  Ex.  11  is  also  represented  by  the  parameter  equations 
4  a;  =  (1  +  «)2,  iy  —  (I  —  t)^.  Test  this  fact  by  substitution,  and  show 
that  the  value  of  dy/dx  obtained  from  these  equations  agrees  with  Jhe 
value  obtained  in  Ex.  11.  [The  curve  is  most  easily  drawn  from  the 
parameter  equations.] 

If  t  denotes  the  time  in  seconds  since  a  particle  moving  on  this  curve 
passed  the  point  (1/4,  1/4),  find  the  total  speed  of  the  particle  at  any 
time.     (See  Ex.  7.) 

30.  Rates.  In  using  the  notation  dy/dx  for  a  derivative, 
we  called  attention  to  the  fact  that  this  symbol  does  not 
represent  a  fraction,  but  rather  the  limit  of  a  fraction ; 
dy/dx  =  lim  Ay/^x. 

We  may,  however,  think  of  any  quantity  as  a  fraction  by 
simply  providing  it  with  a  convenient  denominator;  thus 
3  =  12-7-4,  which  is  a  very  convenient  way  of  writing  3  if  we 
wish  to  add  it  to  1/4. 

In  the  case  of  any  rate  of  change,  it  is  very  usual  to  do  this  ; 


50  ALGEBRAIC  FUNCTIONS  [III,  §  30 

thus  a  speed,  even  though  it  be  thought  of  as  instantaneous,  is 
usually  told  in  feet  per  second,  i.e.  it  is  mentioned  as  if  it 
were  an  average  speed  over  a  whole  second.  A  slope  —  even 
of  a  curve  at  a  point  —  is  spoken  of  as  the  tangent  of  an  angle, 
which,  by  definition,  is  the  ratio  of  one  distance  to  another 
distance.  The  death  rate  in  a  city  or  in  a  state  is  usually 
given  per  100,000  inhabitants,  though  it  is  understood  that  the 
city  does  not  have  exactly  100,000  inhabitants.  Even  the  death 
rate  due  to  a  particular  disease  —  say  appendicitis  —  is  quoted 
per  100,000 ;  the  statement  that  98.4  persons  per  100,000  die 
annually,  does  not  mean  that  98.4  in  any  given  100,000  die,  for 
the  number  of  deaths  is  clearly  an  integer;  the  denominator 
100,000  is  used  solely  for  convenience  and  for  the  purpose  of 
ready  comparison  between  one  city  and  another,  or  between 
one  disease  and  another. 

Rates  are  usually  stated  in  some  such  convenient  manner. 
As  in  the  case  of  death  rate,  such  a  common  denominator  is 
useful  in  all  comparisons  between  different  rates  of  change  of 
the  same  character ;  to  compare  a  speed  of  56  feet  per  second 
with  a  speed  of  40  miles  per  hour  it  is  highly  desirable  to  re- 
duce them  to  a  common  denominator,  and  to  express  both  of 
them,  for  example,  in  feet  per  second. 

31.  The  Differential  Notation.  A  device  of  exactly  this 
character  is  often  convenient  in  our  symbol  for  a  derivative  ; 

if  we  are  dealing,  for  ex- 
ample, with  the  slope  of  a 
curve,  we  have 

(1)  m  =  tan«=lim^  =  ^ 
Ai=oAx      dx 

where  a  is  the  angle  XHT. 
In   this   case    a    convenient 
denominator    is    already   in 
the  figure;   for  in  the  triangle  MPK, 


T 

~~^ 

'A 

K 
M 

.^^<A\ 

0 

11    / 

A 

L       1 

i 

Fig.  12. 


Ill,  §  31]  DIFFERENTIALS  51 

(2)  m  =  tan  a  =  tan  XHT  =  tan  MPK=  ^  =  ^^, 
^  ^  PM      Ax 

where  Aa;  =  PM=  AB. 

This  results  in  throwing  m  into  the  form  of  a  fraction,  with  a 
denominator  A.r,  a  quantity  with  which  we  are  quite  familiar ; 
Ax  means,  as  before,  the  difference  of  any  two  values  of  x,  and 
this  may  be  any  amount  we  desire  except  zero. 

The  new  quantity  MK,  the  height  of  the  triangle  MPK,  is 
called  the  differential  of  y,  and  it  is  denoted  by  the  symbol  dy\ 
its  value  is 

(3)  dy  =  tn^3c, 

which,  varies  for  different  values  of  Ax. 

In  particular,  if  the  curve  is  the  straight  line  y  =  x,  we  find 
111  =  1:,  hence  the  differential  of  x  is 

(4)  dx  =  1  •  Aa?. 

If  we  divide  (3)  by  (4)  we  find 

(5)  dy  -i-dx=  m, 

where  dy-r-dx  now  denotes  a  real  division,  since  dy  and  dx 
are  actual  quantities  defined  by  the  equations  (3)  and  (4),  and 
dx  (=  Ax)  is  not  zero. 

Since  m  stands  for  the  derivative  of  y  with  respect  to  x,  it 
follows  that  that  derivative  is  equal  to  the  quotient  of  dy  by 
dx, 

this  fact  is  the  reason  for  our  use  of  the  symbol  dy/dx  to  repre- 
sent a  derivative  originally. 

In  the  figure  all  quantities  here  mentioned  are  shown: 

dx  =  ^x  =  AB,  dy  =  MK,  A>,  =  MQ,   ^  =  tan  fi,   ^^=tan«. 

Ax  dx 

MK=  dy  is  the  change  that  would  have  taken  place  in  y,  for 
the  change  AB  =  dx  in  x,  if  dy/dx,  the  instantaneous  rate  of 
change  (or  the  slope  at  P),  had  been  maintained.     The  quan- 


52  ALGEBRAIC   FUNCTIONS  [III,  §  31 

titles  dx(=  Ax),*  dy(=:  m  Ax),  Ay,  Ay  —  dy  (=  KQ),  are  infini- 
tesimal when  Ax  approaches  zero,  i.e.  they  approach  zero  as  Ax 
approaches  zero. 

32.  Differential  Formulas.  For  any  given  function  y  =/(x), 
dy  can  be  computed  in  terms  of  dx{=Ax),  by  computing  the 
derivative  and  multiplying  it  by  dx.  Thus,  if  y  =  x\  m  = 
dy/dx  =  2 X,  and  dy  =  mdx  =  2xdx;  again,  it  y  =  x'^—12 x+7, 
m  =  3  x'^  —  12  and  dy  =  m  dx  =  (3  x^  —  12)  rfx. 

Every  formula  for  differentiation  can  therefore  be  written  as 
a  differential  formula ;  the  first  six  in  the  list  in  §  24,  p.  40,  be- 
come after  multiplication  by  dx : 

[I]  dc  =  0.     (The  differential  of  a  constant  is  zero.) 

[II]  d{G  -u)  —c  •  du. 

[III]  d(u -\- v)  =  du  +  dv. 

[IV]  d{u-)  =  nu^-Mu. 
NdD 


[VI]  d(u  ■  v)  =  u  dv  +  V  du. 


Rules  [VII],  [VII J,  of  §  24,  p.  40,  and  [VII^],  of  §  28, 
p.  46,  appear  as  identities,  since  the  derivatives  may  actually 
be  used  as  quotients  of  the  differentials.  From  the  point  of 
view  of  the  differential  notation  Rule  [VII]  merely  shows  that 
we  may  use  algebraic  cancellation  in  products  or  quotients 
which  contain  differentials. 

Rules  [I]-[VI]  are  sufficient  to  express  all  differentials  of 
simple  algebraic  functions.  A  great  advantage  occurs  in  the 
case  of  equations  not  in  explicit  form,  since  all  applications  of 
Rule  [VII]  reduce  to  algebraic  cancellation  of  differentials. 

*  This  equation  does  not  assign  any  particular  value  to  dx  but  only  makes 
it  coincide  with  the  value  of  Ax  chosen  above.  While  we  usually  think  of  an 
infinitesimal  as  small,  because  at  last  it  always  becomes  small,  any  partic- 
ular value  of  an  infinitesimal  is  a  fixed  finite  quantity  and  may  be  chosen  at 
pleasure. 


Ill,  §  32]  DIFFERENTIALS  53 

Example  1.    Given  y  =  x^  —  12  x  +  7,  to  find  dy  and  m. 
dy  =  d(x^  -  12  a;  +  7)  =  d(3fi)  -  d(l2  x)  +  d{l)  =  3  x:^dx  -  12  dx, 
whence      m  =  dy  ^  dx  =  3  x:^  —  12  as  in  Example  3,  p.  10. 

Example  2.   Given  y  =  ^^ — ,  to  find  dy  (Example  1,  p.  33). 

dv  =  ^-^  ^  ~  '^)^^'^^"  +  3)  -  (a;"  +  3)  d(3  X-  7) 
(3x-7)2 

^  (3x-7)-2a;-(x--'  +  3)-3^^^ 
(3x-7)2 

Example  3.    Given  y  =  (x^  +  2)3,  to  find  dy  (Example  1,  p.  36). 
dy  =  dl{x^  +  2)8  j  =S(x-^  +  2)2  d(x^  +  2) 
=  3(x2  +  2)2  .  2  X  •  (Zx. 


Example  4.    Given  y  =  xP  —  2y/Sx-  +  -4,  to  find  dy  (Example  3,  p.  41). 
dy  =  J(x3)  -  2  dv'3x2  +  4 

=  3  x2dx  -  2 ^  .  d(3  x2  +  4) 

2V3x2  +  4 

=  ^3x2 6  X  ]  dx. 

\  V3  x^  +  4       / 

Example  5.    Given  x-  +  y"^  =  1,  to  find  dy  in  terms  of  dx  (§  26,  p.  44). 
d(x2  +  2/2)  =  d(l)  =  0  ;  but  d(x2  +  2/2)  =  d(x2)  +  d(j/2) 
=  2  xdx  +  2  ydy  ; 
hence  2  x  dx  +  2  y  dy  =  0,  or  dy  =  —  (x/y)  dx,  orm  =dy/dx  —  —x/y. 

Example  6.    To  find  dy  and  m  when  x^  +  yS  _  3  xy  =  0  (Example  1, 
p.  46).  d(x3)  +  d(2/3)  -  3  d(xy)  =  0, 

or  3  x2  dx  +  3  2/2  dy  -  3  X  dy  -  3  2/  dx  =  0, 

or  (x2  -y)dx+  (y2  -  x)  dy  =  0, 

whence  dy  =  i^-=^  die,  or  ?n  =  ^  =  ?^fl^. 

y2  —  X  dx      y2  —  x 

Example  7.    To  find  dy  in  terms  of  dx  when  x  —  3  t  +  4,     y  —  (-  -^  2 
(Example  2,  p.  37). 

We  find  dx  =  d(3  «  +  4)  =  3  d< ;  dy  =  d  (t-  +  2)  =2tdt] 

hence  m  =  dy  ^  dx  =  (2/3)  «,  or  dy  =  (2/3)  «  dx  ; 

but  since  f  =  (x  —  4)/3,  this  may  be  written  : 

dy=  (2/9)  (x-4)dx,    or   m  =  ^' =  (2/n)(x  -  4). 
dx 


54  ALGEBRAIC   FUNCTIONS  [III,  §  32 

EXERCISES  XII.  —  DIFFERENTIALS 

[These  exercises  may  be  used  for  further  drill  in  differenti- 
ation, and  for  reviews.  It  is  scarcely  advisable  that  all  of 
them  should  be  solved  on  first  reading.] 

Calculate  the  differentials  of  the  following  expressions  : 
1.    {a)  y  =  a  +  2bx  +  cx"-^.  (6)  y  ={a-\-  x^)^. 

•  (c)   y={a-  bz^y.  (d)  y={a-\-bx-  cx^)^. 


2. 

w''=(''+^)'- 

^'^   ^      (a-bxy 

3. 

(a)  y  =  x2(a  -  x)K 

(c)   2/ =  x*  (a  -  2  x3)2. 

4.  (a)  y  =  v'2  X  +  x'^. 
(c)  J/  =  v^l  -  X*. 

5.  (a)  s  =  tVT+1. 
(c)  2/  -         ' 


V'2  X  +  x'^ 


(ft)  2/  = 

a  +  6x 

(d)  y-- 

1 

(«  +  te  +  CX2)2 

(b)  y-- 

=  (l-2x)(l+3x). 

(d)  (a 

-  6x)2(c  +  dx^). 

(6)   s  = 

=  «a/1  -  <-. 

(.d)  y  = 

=  (a  +  x)  Va  —  X. 

(6)    s. 

=  t'y/a  -  t. 

(d)   s. 

1  -« 
l  +  t 

(6)  9  = 

l-2r2 

9         ^2 

1 


(^)    g=,2  +  2r-3'  ^^  ^         ^8  +  3-^ 


7.    Ca^  v=-- — .  (ft)  2/ =  -7 


^(2-^3)4-  ^(tf-a)3 


8.  (a)  2/=(a +  fta-»)p.  (ft)  y=-^a  +  bxn. 

(c)  2/  = ?-— -.  (d)  2/  =         ^ 

(a  +  6x»)p  ^a  +  ton 

9.  (a).  =  l  +  :^.  (ft)    .=V^- 

l-V?/  ^a-by 


(a  +  6x)3/2 


Ill,  §  32]  DIFFERENTIALS  55 

10.  (a)r=(-V^4-»^)V(5  +  2^.       (,)  ,  ^  ^^  _  ^^^'^^7|y . 

11.  (a)  ^^^'i^.  (i)  g^«-^  +  a«^  +  a«  +  l. 

«  -  1  a  4-  1 

(C)     S  =  «(a2  +  «2)  Va2  _  a2.  ((JT)    e  =  „3(a2  _  „2)3/2 

12.  (a)  y  =  («  +  6x)-i.  (6)  ?/  =  (a  +  6x)-2. 
(c)  y  =(a  +  bx^)-K                      (d)  y  ={a  +  bx^)-^. 

X3.(„).=(ii|i:y.       (^)"(f^:)-- 

(c)   z  =(Ay-3  +  ^y-5)2.  (rf)  z  ={Ay-^  +  By-^y\ 

(e)  y  =  A{a  +  bxy°  +  B{a  -  ftx)-i". 

14.  Determine  dy  in  terms  of  dx  from  the  equations  below  : 

(a)  Ax  +  By  +  C  -0.  (g)   y/x  +  Vy  =  c. 

X(6)   xy  +  y  =  l.  (h)  (l-ax)(x^  +  y^)  =  i. 

\,^c)    x^-2xy-3y^  =  0.  (i)   x^  +  y^  =  {ax  +  by. 

x2  _  y2  U;    y.2      a-bx 

(e)    j^2(x  -  ffl)  =  (x  +  af.  (k)  y^  -  5  axy  +  x^  =  0. 

(/)  y*-2y^x-l  =  0.  (Z)    (X  +  y)3/2  +  (X  -  2/)3/2  =  a3/2. 

15.  Obtain  the  equation  of  the  tangent  at  (2,-1)  to  the  curve 

4  x'  -  2  xj/  -  5  y-  _  6  X  -  4  ?/  -  7  =0. 

16.  Obtain  the  equation  of  the  tangent  at  (2,  1)  to  the  curve 
/        x3  -  7  x^j/  -  5  2/3  +  4  x2  -  10  xy  +  8  X  —  5  2/  +  18  =  0. 

17.  Obtain  the  equation  of  the  tangent  at  (xo,  j/o)  to  each  of  the  fol- 
lovfing  curves : 

Curve  Tangent 

(a)  y-  =Aax;  yyo  =  2 a(x  +  Xo). 

(6)  x2  +  2/2  r=  a2 ;  xxo  +  yyo  =  cfi- 


<=)S-S-^ 


a^o  I  yyo  _  I 

C2  62 


(d)   (X  +  2/)2  =  1  ;  (X  +  y){xo  +  J^o)  =  1- 


56  ALGEBRAIC   FUNCTIONS  [III,  §  32 

18.    Find  the  derivative  dyjdx  for  the  curves  defined  by  each  of  the 
pairs  of  parameter  equations  given  below : 

l  +  «  I         3^^^'  at 

1  +  «  1^2'  i  ^  a2«2 

1 
4  7rr-' 


(d) 


(« 

[  a;  =  4  7rr-2, 

(e) 

i.  =  |... 

e-2.  r       3  47rr3 


19.  If  a  particle  moves  so  that  its  coordinates  (a;,  y)  at  any  time  t  are 

"^  ~  1  +  <2'     ^      1  +  i2' 

show  on  the  same  diagram  the  values  of  x  and  of  y  in  terms  of  time ; 
what  are  the  extreme  values  of  x  and  of  ?/,  and  when  are  they  attained  ? 
From  the  diagram  construct  another  showing  the  (x,  y)  curve  followed 
by  the  particle. 

20.  Calculate  the  x  and  y  components  of  the  speed  {v^  and  Vy)  at  any 
time  «,  and  the  resultant  speed  Vi'x-  +  Vy^^  along  the  path,  in  the  motion 
of  Ex.  19.     Show  that  Uj,  -=-  Vx  =  dy/dx.     See  Ex.  7,  p.  49. 

21.  If  a  particle  moves  so  that  its  coordinates  in  terms  of  the  time  are 

X  =  1  -  «  +  <^     y  =  1  +t  +  t-, 
show  that  its  path  is  a  parabola.     Show  that  from  the  moment  «  =  0  its 
speed  steadily  increases. 

22.  A  point  moves  on  a  straight  line  so  that  its  distance  s  from  a  fixed 
point  on  the  line  at  any  time  t  is  as  given  below.  Describe  the  motion 
from  «  =  0,  giving  the  times  when  the  speed  is  positive,  negative,  zero. 
Draw  the  (s,  t)  diagrams  and  the  {v,  t)  diagrams. 

(a)  s  =  «2-4<  +  3. 

(&)  s  =  «3  -  15  «2  +  68 «  -  4. 

(c)  s  =  3<*  -  40«3  +  54  «2  _  10;  ^ 

23.  If  the  volume  of  a  sphere  increases  at  the  rate  of  2  cu.  ft.  a  second, 
calculate  the  rate  of  change  per  second  of  the  radius  and  of  the  surface. 
What  are  these  rates  when  the  volume  is  100  cu.  ft.  ? 

24.  If  B  denotes  the  radius  of  a  sphere,  S  the  surface,  and  V  the 
volume,  calculate  the  differential  of  each  of  these  in  terms  of  each  of  the 
othei-s. 


Ill,  §  32]  DIFFERENTIALS  57 

25.  If  the  radius  of  a  cylinder  expands  at  the  rate  of  1/2  in.  a  second, 
starting  with  a  value  5  in.,  and  if  the  height  remains  fixed  at  10  in.,  at 
what  rate  per  second  is  the  volume  changing  at  any  time  t  ?  When 
t  —  10  sec.  ?    The  same  for  the  total  surface  ? 

^ ,  26.  When  you  walk  straight  away  from  a  street  lamp  with  uniform 
speed,  does  the  end  of  your  shadow  also  move  with  uniform  speed  '.' 
Supposing  that  your  height  is  70  in.,  show  how  fast  the  shadow  tip  moves 
if  you  walk  6  ft.  per  second  away  from  a  lamp  10  "ft.  above  ground. 

27.  The  electrical  resistance  of  a  platinum  wire  varies  with  the  tem- 
perature, according  to  the  equation 

B  =  Eo(l-ae  +  bd-)-^; 
calculate  dB  in  terms  of  d0.     What  is  the  meaning  of  dR/dd  ? 

28.  Van  der  Waal's  equation  giving  the  relation  between  the  pressure 
and  volume  of  a  gas  at  constant  temperature  is 

Draw  the  graph  when  a  =  .0087,  b  =  .0023,  c  =  1.1.  Express  dv  in  terms 
of  dp.    What  is  the  meaning  of  dv/dp  ? 

29.  The  crushing  strength  of  a  hollow  cast  iron  column  of  length  I, 
inner  diameter  d,  and  outer  diameter  D,  is 


r=46. 


Vo(^^ 


Calculate  the  rate  of  change  of  T  with  respect  to  D,  d,  and  I,  when  each 
of  these  alone  varies. 

30.  Show  that  the  curve  y  =  (x—  ay  +  b  has  no  maxima  or  minima. 

31.  Proceed  as  in  Ex.  30  for  y  =  (x  —  ay  +  b. 

32.  Show  (see  Exs.  10-14,  p.  38)  that  the  curve  y  =  P(x)  is  tan- 
gent to  the  X-axis  at  points  where  the  polynomial  P(x)  has  a  double  root. 

33.  Show  that  if  P{x)  is  a  polynomial,  its  double  roots  are  also  roots 
of  the  polynomial  P'{x)  =  dP(x)/dx.  Hence  the  H.  C.  D.  of  P(x)  and 
P'(x)  contains  as  a  factor  x—  k,  where  k  is  the  double  root. 

34.  Assuming  the  principle  of  Ex.  33,  find  the  double  roots  of  each  of 
the  following  equations : 

(a)  x3  +  a:2  -  5x  +  3  =  0.  (f)    x*  +  2  x^  -  11  x2-12  x  -|-  30  =  0. 

(6)  x'^  +  3  x2  -  4  =  0.  (J)  X*  -h  2  x3  -  2  X  -  1  =  0. 


CHAPTER   IV 
FIRST   APPLICATIONS   OF   DIFFERENTIATION 

PART   I.     APPLICATIONS   TO   CURVES  — EXTREMi:S 

33.   Tangents  and  Normals.     We  have  seen  in  §  4,  p.  6,  that 
if  the  equation  of  a  curve  C  is  given  in  explicit  form : 

(1)  y=f{^), 

the  derivative  at  any  point  P  on  C  represents  the  rate  of  rise, 
or  slope,  of  (7  at  P: 

^^^    [S  ati-^t^'^^^'  ofC]^,j.  =  slo2?e  ofPT=tan  a=[m]  ^tP, 

where  a  is  the  angle  XIIT,  counted  from  the  positive  direction 
of  the  X-axis  to  the  tangent  PT,  and  where  m^  denotes  the 

slope  of  C  at  P. 

Hence  (§  4,  p.  7)  the  equation  of 

the  tangent  is 

where  the  subscript  P  indicates  that 
the  quantity  affected  is  taken  with 
the  value  which  it  has  at  P. 
If  the  slope  nip  is  positive,  the  curve  is  risi7ig  at  P  ;  if  lUp  is 
negative,  the  curve  is  falling  ;  if  mp  is  zero,  the  tangent  is  hori- 
zontal (§  6,  p.  8).  Points  where  the  slope  has  any  desired 
value  can  be  found  by  setting  the  derivative  equal  to  the  given 
number,  and  solving  the  resulting  equation  for  x. 


Y 

\ 

V 

^T 

.^ 

^ 

X 

0 

^U 

AN\ 

IV,  §  34]  TANGENTS  AND   NORMALS  59 

Since,  by  analytic  geometry,  the  slope  n  of  the  normal  PN 
is  the  negative  reciprocal  of  the  slope  of  the  tangent,  we  have, 

(4)  n,  =  slope  ofPN=-^=-        )        , 

as  in  Ex.  8,  p.  11,  hence  the  equation  of  the  normal  is : 

34.    Tangents  and  Normals  for  Curves  not  in  Explicit  Form. 

The  equation  of  the  curve  may  not  be  given  in  the  explicit 
form  (1);  instead,  it  may  not  be  solved  for  either  letter 

(1)  F(x,y)=0, 

as  in  §§  26-27,  pp.  44-45;  or  it  may  be  solved  for  x: 

(2)  x  =  <l>(y), 

as  in  §  28,  p.  46 ;  or  the  equations  in  parameter  form  may  be 
given : 

(3)  x=f((),  y  =  <i>{t), 
as  in  §  29,  p.  47. 

In  any  of  these  cases,  chj/dx  can  be  found  by  the  methods 
of  the  articles  just  cited,  and  this  value  may  be  used  in  the 
formulas  of  §  33.     No  new  formulas  are  necessary. 

In  the  particular  case  of  the  parameter  form  (3),  however,  a  special 
formula  is  sometimes  useful.     Since  by  §  29, 

the  equation  of  the  tangent  becomes,  after  simplification, 

^  LdtAp  ^  LdtSp'         y-Vp        z-Zp 

and  the  equation  of  the  normal  is 

A  special  formula  for  equations  in  the  implicit  form  (1)  will  be  given 
later  (§  164) ;  just  now  it  would  actually  be  inconvenient. 


60         APPLICATIONS  OF  DIFFERENTIATION      [IV,  §  35 

35.  Secondary  Quantities,     in  Fig.  13,  §  33,  since 

tan  «  (=  tiip  =  [rfy/dx]p),  and  AP(  =  yp), 
are  supposed  to  be  known,  the  right  triangles  HAP  and  PAN'  can  both 
be  solved  by  the  rules  of  Trigonometry,  and  the  lengths  HA,  AN",  HP, 
PiV  can  be  found  in  terms  of  mp  and  yp : 

[Subtangentjp  =^J.  =AP-i-  tan  a  =  yp-^mp=  [y/m]p. 
[Subnormal]/.  =  AN  =  AP  •  tan  «  =  [?/.  m]/>,  since  a  =  Z  APX. 
[Length  of  tangent]/.  =  HP  =  ^AP'^  +  HI^  =  Vy%  +  [2//m]^ 

=  [y  Vl  +  (l/m)^]p 
[Length  of  normal]/.  =  P.V  =  ^AP^  +  AN'  =  \/y],  +  (y  •  m)2, 

=  ly  Vi  +  TO-^]p. 

It  is  usual  to  give  these  lengths  the  names  indicated  above  ;  and  to  cal- 
culate the  numerical  magnitudes  of  these  quantities  without  regard  to 
signs,  unless  the  contrary  is  explicitly  stated. 

36.  Illustrative  Examples.  In  this  article,  a  few  typical 
examples  are  solved. 

Example  1.  Given  the  curve  y  =  x^—12x  +  7  (Ex.  2,  p.  25) ,  we  have 
m  =  dy/dx  =  3x^—12. 

(1)  The  tangent  (T)  and  the  normal  (N)  at  a  point  where  x  =  a  are 

(T)  y-  (a3-12a  +  7)  =  (3a2-12)  (a;  -  a), 

(,V)  y_(a8_12a  +  7)=^-=^(x-a); 

thus,  at  a;  =  3,  the  tangent  and  normal  are 

(T)  y  +  2  =  15(x-3),     (iV)  y  +  2  =  -j-V(x-3). 

(2)  The  tangent  has  a  given  slope  k  at  points  where 


3x2-12  =  A;  -•-    "-    '  -'^'  +  12 


there  are  always  two  points  where  the  slope  is  the  same,  if  i  >  —  12 ; 
thus  if  A;=0,  x=±2;  il  k  =  —9,  x=±l  ;  if  A;  =-12,  x=0 ;  if  ^•<  — 12, 
no  real  value  for  x  exists  (see  Fig.  17,  p.  77). 

(3)  The  secondary  quantities  of  §  35  may  be  calculated  without  using 
the  formulas  of  §  35.  Thus,  at  the  point  where  x  =  3,  the  tangent  (T) 
cuts  the  X-axis  where  x  =  47/15  ;  the  normal  (N)  cuts  the  x-axis  where 
X  =  —  27.     If  the  student  will  draw  a  figure  showing  these  points  and 


IV,  §  36]  TANGENTS  AND   NORMALS  61 

lines,  he  will  observe  directly  that  the  subtangent  is  2/15,  the  subnormal 
30,  the  length  of  the  tangent  y/W+lY/Wf,  the  length  of  the  normal 
VSO-  +  22.     These  values  agree  with  those  given  by  §  35. 

(4)  The  given  curve  cuts  the  curve  y  =  a;^  —  5  at  a  point  given  by  solv- 
ing the  two  etiuations  simultaneously  ;  this  gives  a;  =  l,y  =  — 4;  at  this 
point  the  slopes  of  the  two  curves  are  m'l  =  —  9,  jjio  =  +  3  ;  hence,  by 
Analytic  Geometry,  the  acute  angle  between  them  is  given  by  the  formula 

tan^=:"'^-»^-^^--^^^L2^1, 
1  +  miTO.2       1-27       26      13 

from  which  0  can  be  found  by  use  of  a  trigonometric  table  ( Tables,  V,  A). 
From  a  larger  table,  we  find  6  =  24^  47'. 

Example  2.  Given  the  circle  x"^  -\-  y-  —  \,  we  have  m  =  dy/dx  =  —  x/y 
[see  §  26]. 

(1)  The  tangent  (T")  and  normal  (iV)  at  a  point  (xo,  yo)  are 

IT)  (y-y,)  =  -^(x-Xo),     (X)(y-yo)^y^(x-xo); 
yo  xo 

or,  since  Xq^  +  j/q^  =:  1, 

{T)  xxo  +  2/1/0  =  1,     (^")  y^o  =  yox  ; 
thus,  at  the  point  (3/5,  4/5),  which  lies  on  the  circle,  we  have 

(r)3a;  +  4y  =  5,     (N)3y  =  4x. 

(2)  The  tangent  has  a  given  slope  k  at  points  where 

-^  =  k,    i.e.    xo  +  ktjo  =  0. 

2/0 

The  coordinates  (xo,  2/o)  can  be  found  by  solving  this  equation  simul- 
taneously with  the  equation  of  the  circle,  or  by  actually  drawing  the  line 
Xo  +  kyo=0.  Thus  the  points  where  the  slope  is  +  1  lie  on  the  straight  line 
a;  4-  2/  =  0  ;  hence,  solving  x  +  2/  =  0  and  x-  +  y'^  =  1,  the  coordinates  are 
found  to  be  X  =  ±  1/ v'2,  y  =  ^  1/ V2  ;  but  these  points  are  most  readily 
located  in  a  figure  by  actually  drawing  the  line  z  +  y  =  0. 

(3)  The  given  circle  cuts  the  parabola  Qy  —  20x2  at  the  points  (±  3/5, 
4/6) ;  at  the  point  (3/5,  4/6)  the  slopes  of  the  two  curves  are  mi  =  —  3/4, 
ma  =  40  x/9  =  8/3  ;  hence  the  acute  angle  6  between  the  two  curves  at 
that  point  is 

tan  e  =  "'^  ~  "^-  =~^  3.4167,  whence  6  =  73°  41'  10". 

1  +  ??li??l2        12 


62  APPLICATIONS  OF  DIFFERENTIATION      [IV,  §  36 

EXERCISES  Xm.  —  TANGENTS  AND  NORMALS 

1.  Find  the  equations  of  the  tangent  and  that  of  the  normal,  and  find 
the  four  quantities  defined  in  §  35,  for  each  of  the  following  curves  at  the 
point  indicated  : 

(a)  7j  =  x^-12x  +  '!;  (1,  -  4).  (e)  x  =  y^  -  3y'^  +  5  ;   (3,  1). 

fb)  y=^^~^  ■   (-1,  3).  (/)    l*^*^^"^^'^!  ;  (1,  1). 

(c)  9x2  +  2/2  =  25;  (1,4).  ,  .    \x  =  t-^+4.t-\\  .   .^  ^  j. 

(d)  X2/  +  ?/2_2x  =  5;   (-4,  1).  ^^^    j?/ =  «' -  3  «  +  5  j      ^         ^' 

2.  Find  the  angle  between  the  curves  y  =  x^  and  y^  =  zat  each  of  their 
common  points.     See  Tables,  III,  A. 

3.  Find  the  points  (if  any)  at  which  each  of  the  curves  in  Ex.  1  has 
the  slope  zero  ;  the  slope  +  1. 

4.  Determine  the  values  of  x  for  which  the  slope,  in  each  of  the  curves 
in  Ex.  1,  is  positive  ;  and  those  for  which  it  is  negative. 

5.  In  Ex.  1,  the  curves  (a)  and  (c)  pass  through  the  point  (1,  —  4); 
at  what  angle  do  they  cross  ? 

6.  The  curves  y  =  x,  y  =  x^,  y  =  x^,  ■•.,  y  =  x~^,  y  =  x~^,  •••,  y  —  x^'^, 
y  =  x^/^,  •••  all  pass  through  the  point  (1,  1).  Determine  the  angle  which 
each  of  these  curves  makes  with  the  first  one  of  them  at  that  point. 

7.  Determine  the  angle  between  the  curves  y  =  x^  and  y  =  x™  at  the 
point  (1,  1)  where  m  and  n  have  any  values  whatever  ;  at  the  point  (0,  0) 
(only  if  both  n  and  m  are  positive).  (Special  case:  n=p/q,  m  =  q/p, 
where  p  and  q  are  integers.) 

8.  Determine  the  equation  of  the  tangent  and  that  of  the  normal  to 
the  ellipse  b^x^  +  a'^y^  =  a%'^  at  any  point  (xo,  yo)  on  it. 

[Solution  :  2  b'^xdx+  2  a^y  dy  =  0,  hence  dy/d3P=—  b'^x/a-y  ;  the  tan- 
gent and  normal  are,  respectively, 

(T)  (y-yo)  =  -^(x-xo),(^)  (y-yo)=^'(x-x,), 

a-yo  0% 

or        (T)  b"xxQ  +  a^yyo  =  a^ft^,  (iv")  b-x^y  —  a-xy^,  =  xoyo^b'^  —  a-), 
since  6%2  +  (^a^^a  _  (^252.] 

9.  Determine  the  equation  of  the  tangent  and  that  of  the  normal  to 
each  of  the  following  curves  at  any  point  (Xq,  j/q)  on  it : 

(a)  y  =  A:x2.  (e)   b^x^  -  aY  =  a^b\ 

(b)  y2  =  2  px.  (/)  ax2  +  2bxy  +  cy^  =  f. 

(c)  x^  +  y^  =  a'\  (g)  ax^  +  2bxy  +  cy"^  +  2dz  +  2  ey +f  =  i 
Id)  y  =  kx\  Qi)  y  =  (ax  +  b)/{cx  +  d). 


IV,  §  38]  EXTREMES  63 

10.  The  curve  whose  equations  in  parameter  form  are 
(1)  x  =  3t  +  4,  y  =  t^  +  2, 

gives  (Example  2,  p.  37): 

dt       dt       3 
hence  this  curve  has  a  slope  1  when  t  =  3/2,  i.e.  when  x  =  17/2,  y  =  17/4. 
Its  slope  is  0  when  t  =  0,  i.e.  at  (4,  2). 

Verify  these  facts  by  drawing  an  accurate  figure ;  also  by  eliminating 
t  in  (1)  and  finding  the  derivative  from  the  explicit  equation. 

11.  Show  that  the  slope  of  the  curve  z^  +  y^  —  Sxy  =  0  (Example  1, 
p.  46)  is  +  1  at  points  where  it  cuts  the  circle  x-  +  y-  —  x  —  y  =  0. 
Show  that  its  slope  is  zero  (tangent  horizontal)  where  it  cuts  the  parabola 
y  =  a;2  ;  that  the  tangent  is  vertical  (1/m  =  0)  where  it  cuts  the  parabola 
y2  =x. 

12.  Draw  the  curve  of  Ex.  11  by  using  its  equations  in  parameter  form 
(Ex.  6  d,  p.  48)  : 

1  +  «3'    ^1+^3' 

and  show  that  dy/dx  =(2t  —  ?*)/(!  —  2  «'),  found  from  these  equations, 
agrees  with  the  value  found  from  the  implicit  equation. 

37.  Extremes.  In  §  6,  p.  8,  and  in  numerous  examples, 
we  have  found  maxima  and  minima  of  functions  by  first 
finding  the  points  at  which  the  tangent  is  horizontal,  and 
then  testing  these  values. 

The  value  of /(x)  at  a  point  where  x=a  is /(a).    This  value  is 

r  maximum  1      ,       -^   -^    ■    f  greater  than  ]  ^,  , 

a  ^      .   .  \  value  it    it    is  K       ^,  any   other   value 

[  minimum  J  [  less  than        j      *' 

of /(x")  for  values  of  x  sufficiently  near  to  re  =  a. 

A  maximum  or  a  minimum  is  called  an  extreme  value,  or  an 

extreme  oif(x). 

38.  Critical  Values.  We  have  seen  that  a  horizontal  tangent 
(i.e.  slope  zero)  does  not  always  give  rise  to  an  extreme. 
Thus,  the  curve  y  =  x^  (Ex.  5(6),  p.  11)  has  a  horizontal  tangent 
at  the  origin ;  but  the  origin  is  neither  a  highest  nor  a  lowest 
point. 


64         APPLICATIONS  OF  DIFFERENTIATION     [IV,  §  38 

On  the  other  hand,  extremes  may  also  occur  at  points  where 
the  derivative  has  no  meaning,  or  at  points  where  the  function 
becomes  meaningless. 

Thus,  the  curve  y  =  x-V3  gives  m  =  2/(3  x^s)  ;  hence  m  is  meaningless 
when  X  =  0  ;  in  fact,  the  curve  has  a  vertical  tangent  at  that  point.     It  is 

easy  to  see  that  this  is, 
however,  the  lowest  point 
on  the  curve. 

Again,  if  a  duplicating 
apparatus  costs  $  150,  and 
if  the  running  expenses  are 
1  (*    per    sheet,    the    total 
Fig.  14.  ^^^^   ^^  printing  n  sheets 

is  «  =  150  +  .01  n.  This  equation  represents  a  straight  line  ;  geometri- 
cally there  are  no  extreme  values  of  t ;  but  practically  t  is  a  minimum 
when  n  =  0,  since  negative  values  of  n  are  meaningless.  Such  cases  are 
usually  easy  to  observe. 

A  value  of  x  of  any  one  of  the  types  just  mentioned,  at  which 
f(x)  may  have  an  extreme,  is  called  a  critical  value;  the  cor- 
responding point  on  the  curve  y=f{^)  is  called  a  critical 
point. 

39.  Fundamental  Theorem.  We  proceed  to  show  that  a  func- 
tion f{x)  cannot  have  an  extreme  except  at  a  critical  point : 
that  is,  assuming  that  f{x)  and  its  derivative  have  definite 
meanings  at  x  =  a  and  everywhere  near  x  =  a,no  extreme  can 
occur  if  the  derivative  is  not  zero  at  x  =  a. 

We  are  supposing  that  all  our  functions  are  continuous ;  if, 
then,  the  derivative  m  is  positive  at  a;  =  a,  it  cannot  suddenly  ^ 
become  negative  or  zero.     Hence  m  is  positive  on  both  sides  of 
x  =  a,  and  there  can  be  no  extreme  there. 

Likewise  if  m  is  negative,  the  curve  is  falling  near  x  =  a 
on  both  sides  of  a;  =  a;  there  can  be  no  extreme. 

40.  Final  Tests.  It  is  not  certain  that  f(x)  has  an  extreme 
value  at  a  critical  point.  To  decide  the  matter,  we  proceed  to 
determine  whether  the  curve  rises  or  falls  to  the  left  and  to 


IV,  §  41] 


EXTREMES 


65 


the  right  of  the  critical  point:  it  rises  if  m>0;  it  falls  if 
m  <  0. 

Near  a  maximum,  the  curve  vises  on  the  left  and  falls  on  the 
right. 

Near  a  minimum,  the  curve  falls  on  the  left  and  rises  on  the 
right. 

If  the  curve  rises  on  both  sides,  or  falls  on  both  sides,  of  the 
critical  point,  there  is  no  extreme  at  that  point. 

41.   Illustrative  Examples. 

Example  1.  To  find  the  extremes  of  the  function  y=f(x)  =  r'^— 12  x-f-7. 
(See  §  6,  p.  10.) 

{A)  To  find  the  Critical  Values.  Set  the  derivative  equal  to  zero  and 
solve  for  x : 

^  _  f'.v . 


dx 


:  3  a:2  -  12  ;  3  x^  -  12  =  0  ;  x  =  2  or  x 


(JB)  Precautions.    Notice  that/(x)  and  its  derivative  each  has  a  mean- 
ing for  every  value  of  x  ;  hence  x  =  +  2  and 
X  =—  2  are  the  only  critical  values. 

(C)  FinalTests.  m=3x2-12=3(x2-4) 
is  positive  if  x  is  greater  than  2,  nega- 
tive if  X  is  slightly  less  than  2  ;  hence  the 
curve  rises  on  the  right  and  falls  on  the  left 
of  X  =  2,  therefore /(2)  =  —  9  is  a  minimum 
of  /(x).  The  student  may  show  that 
/(— 2)  =23  is  a  maximum  of /(x).  (See 
Fig.  5,  p.  10.) 

Example  2.  To  find  the  extremes  of  the 
function 

2/=/(x)=3x*-12x8  +  50. 

{A)  Critical  Values.  Setting  f?2//f?x  =  0, 
and  .solving,  we  find  : 


^=12x8. 
dx 


12  x8  -  36  x2 


X  =  0,  or  X  =  3. 
{B)  Precautions,    y  and  dy/dx  have  a 
meaning    everywhere ;    the    only    critical 
values  are  0  and  3. 


1 

\ 

\ 

'\ 

\ 

] 

1 

2/ 

= 

X 

- 

IL 

z|f+5(jj 

LL 

1  2 

i 

1 

I  i 

Fig.  15. 


66         APPLICATIONS  OF  DIFFERENTIATION     [IV,  §  41 

(C)  Final  Tests.  Near  x  =  0,  m  =  12x-(x  —  3)  is  negative  on  both 
sides  ;  hence  there  is  no  extreme  there,  though  the  tangent  is  horizontal. 

Near  x  =  3,  m  =  12  x^{x  —  3)  is  positive  on  the  right,  negative  on  the 
left ;  hence  / (3)  =  —  31  is  a  minimum. 

The  information  given  above  is  of  great  assistance  in  accurate  drawing. 

Example  3.  Tvfo  railroad  tracks  cross  at  right  angles  ;  on  one  of  them 
an  eastbound  train  going  15  mi.  per  hour 
clears  the  crossing  one  minute  before  the 
engine  of  a  southbound  train  running  at 
20  mi.  per  hour  reaches  the  crossing.  Find 
vyhen  the  trains  were  closest  together. 

Let  X  and  y  be  the  distance  in  miles  of 
the  rear  end  of  the  first  train  and  the  en- 
gine of  the  second  one  from  the  crossing, 
respectively,  at  a  time  t  measured  in  min- 
utes beginning  with  the  instant  the  first 
train  clears  the  crossing  ;   then 

60  60^  "        <J     9        144 

where  D  is  the  distance  between  the  trains  in  miles. 

Since  D  is  a  positive  quantity,  it  is  a  miniimini  whenever  D-  is  a  mini- 
mum ;  hence  we  write  : 

^  =  ^^!i  =  _2  +  25         _2^25^^  ^^16 

dt  9      72    '         9      72  '  25' 

when  «<  16/25,  m<0;  if  <>  16/25,  m>0;  hence  D^  is  diminishing 
before  t  =  16/25  and  increasing  afterwards.  It  follows  that  Z)  is  a  mini- 
mum when  t  =  16/25.     Substituting  this  value  for  t,  we  find 

25  25  25 

hence  the  minimum  distance  between  the  trains  is  1/5  of  a  mile,  and  this 
occurs  16/25  of  a  minute  after  the  first  train  clears  the  crossing. 

Example  4.  To  find  the  most  economical  shape  for  a  pan  with  a  square 
bottom  and  vertical  sides,  if  it  is  to  hold  4  cu.  ft. 

Let  X  be  the  length  of  one  side  of  the  base,  and  let  h  be  the  height.  Let 
V  be  the  volume  and  A  the  total  area.  Then  V  =  hx?  =  4,  whence 
h  =  4/x2;  and  -.n, 

A  =  x'^  +  4:hx  =  x^  +  —  ; 

X 


IV,  §  41]  EXTREMES  67 

whence  we  find 

dx  x:^  x? 

When  a;  <  2,  m  =  2{x?  —  ^)lx^  is  negative;  when  x>2,  m  is  positive; 
hence  A  is  decreasing  when  x  is  increasing  toward  2,  and  A  is  increasing 
as  X  is  increasing  past  2  ;  therefore  x  —  2  gives  the  minimum  total  area 
yl  =  12.  Notice  that  the  height  is  A  =  4/x2  =  1.  The  correct  dimensions 
are  x  =  2,  /i  =  1  (in  feet) . 

Example  5.  The  pan  of  the  preceding  example  is  to  sit  under  a 
refrigerator  which  clears  the  floor  by  8  in.     How  should  it  be  made  ? 

Since  h  cannot  now  exceed  8  in.  =  2/3  ft.,  it  is  clear  that  the  mini- 
mum of  .4  found  in  Ex.  4  does  not  apply.  The  function  A  =  x'^  -\-  IG/x 
is  meaningless  if  A  >  2/3,  i.e.  if  4/x'^>2/3,  or  x-/4<3/2,  orx<  \/6 
=  2.45  (in  feet). 

Since  A  is  increasing  as  x  increases,  x  should  be  made  as  small  as  pos- 
sible ;  practically,  we  ought  to  chose,  say  x  =  2.5  ft.  =  30  in.  ;  then  h  = 
16/25  ft.  =7.68  in.,  —  we  ought  to  take  h  about  7  3/4  in.,  which  gives 
1/4  in.  clearance.  This  gives  F=  6975  cu.  in.,  in  place  of  6912  required, 
but  this  difference  is  on  the  safe  side,  and  is  practically  negligible,  because 
it  corresponds  to  a  difference  in  height  of  much  less  than  1/8  in, 

EXERCISES  XIV.— EXTREMES 

1.  Determine  the  maximum  and  minimum  values  of  the  following 
functions  and  draw  the  graphs,  choosing  suitable  scales  : 

(a)  2/  =  x3  -  3  x2  +  1.  (ft)  s  =  2  i'i  -  3  <2  -  36  «  +  20. 

(c)  p  =  g^  +  6  ^2  _  15  q^  (f^)  y  =  x3  -  2  ax^  +  a2x. 

(e)  X  =  ?/*  -  8  >/2  +  2.  (/)  r  =  w4  _  4  m3  +  4  „2  +  3. 

{g)  m  =  7i5  _  10  ,^4  +  20  ?i3  +  32.  {h)  ^  =  j-6-6r*  +  4  r"'+9r2-12r+4. 

(0  s  =  ((-l)(2-0^-  U)  V=h(h-1)^. 

(k)  r  =  (s2  -  l)(s2  _  4).  (0  X  =  (2/  -  2)3  (y  +  3)8. 

(.r  +  2)-^ 

(0)  y-         "^ 


x2  +  2  X  +  4 

X3  -  X 


^'^'        X^-X^  +  l 

(s)  Z»  =  rV2-j-2. 


(«)    V  = 

dp)     K: 

h 

ah-  +  bh  +  c 

00     Q  = 

-.k  +  Vl-  k. 

(0  B  = 

:  -^.7+15  -  X. 

68         APPLICATIONS  OF  DIFFERENTIATION     [IV,  §  41 

2.  What  is  the  largest  rectangular  area  that  can  be  inclosed  by  a  line 
100  feet  long  ? 

3.  What  must  be  the  ratio  of  the  sides  of  a  right  triangle  to  make  its 
area  a  maximum,  if  the  hypothenuse  is  constant  ? 

4.  Determine  two  possible  numbers  whose  product  is  a  maximum  if 
the  sum  of  their  squares  is  50.     Is  there  any  minimum  ? 

5.  Determine  two  numbers  whose  product  is  100  and  such  that  the 
sum  of  their  squares  is  a  minimum.  Is  there  any  maximum  ?  Did  you 
account  for  negative  possible  values  of  the  two  numbers  ? 

6.  What  are  the  most  economical  proportions  for  a  cylindrical  can  ? 
Is  there  any  most  extravagant  type  ?  Mention  other  considerations  which 
affect  the  actual  design  of  a  tomato  can.  Is  an  ordinary  flour  barrel 
this  shape  ?     What  different  considerations  enter  in  making  a  barrel  ? 

7.  What  are  the  most  economical  proportions  for  a  cylindrical  pint 
cup  ?     (1  pint  =  28|  cu.  in.)     Mention  considerations  of  practical  design. 

8.  Determine  the  best  proportions  for  a  square  tank  with  vertical 
sides,  without  a  top.    Is  there  any  most  extravagant  shape  ? 

9.  The  strength  of  a  rectangular  beam  varies  as  the  product  of  the 
breadth  by  the  square  of  the  depth.  What  is  the  form  of  the  strongest 
beam  that  can  be  cut  from  a  given  circular  log  ?  Mention  some  other 
practical  considerations  which  affect  actual  sawing  of  timber. 

10.  The  stiffness  of  a  rectangular  beam  varies  as  the  product  of  the 
breadth  by  the  cube  of  the  depth.  What  are  the  dimensions  of  the  stiffest 
beam  that  can  be  cut  from  a  circular  log  ? 

11.  Is  a  beam  of  the  commercial  size  3"  x  8"  stronger  (or  stiffer)  than 
the  size  2"  x  12"  (1)  when  on  edge,  (2)  when  lying  flat? 

[Commercial  sizes  of  lumber  are  always  a  little  short.] 

12.  What  line  through  the  point  (3,  4)  will  form  the  smallest  triangle 
with  the  coordinate  axes  ?   Is  there  any  other  minimum  ?   Any  maximum  ? 

13.  Determine  the  shortest  distance  from  the  point  (0,  3)  to  a  point 
on  the  hyperbola  x"^  —  y^  =  16.     Show  that  it  is  measured  on  the  normal. 

[Hint.    Use  the  square  of  the  distance.] 

14.  The  distance  D  from  the  point  (2,  0)  to  the  circle  x^  -\-y^  =  \  is 
given  by  the  equation  D'^  =  5  —  4  x.  Discover  the  maximum  and  mini- 
mum values  of  D-,  and  show  why  the  ordinary  rule  fails. 

15.  Show  that  the  maximum  and  minimum  on  the  cubic  y  =  x?  —  ax  +  6 
are  at  equal  distances  from  the  y  axis.     Compute  y  at  these  points. 


IV,  §  41]  EXTREMES  69 

16.  Show  that  the  cubic  x^  -  ax  +  h  =  0  has  three  real  roots  if  the  ex- 
treme values  of  the  left-hand  side  (Ex.  15)  have  different  signs.  Express 
this  condition  algebraically  by  an  inequality  which  states  that  the  product 
of  the  two  extreme  values  is  negative. 

[Any  cubic  cau  be  reduced  to  this  form  by  the  substitution  a:  =  3;' +  A ; 
heuce  this  test  may  be  applied  to  any  cubic] 

17.  Show  that  if  the  equation  x^—  ax  +  b  =0  has  two  real  roots,  the 
derivative  of  the  left-hand  side  (i.e.  3  x^  -  «)  must  vanish  somewhere 
between  the  two  roots.     Show  that  the  converse  is  not  true. 

18.  The  line  y  =  mx  passes  through  the  origin  for  any  value  of  m. 
The  points  (1,  2.4),  (3,  7.6),  (10,  2.5)  do  not  lie  on  any  one  such  line : 
the  values  of  y  found  from  the  equation  y  =  mx  at  x  =  1,  S,  10  are  m, 
3  m,  10  m ;  the  differences  between  these  and  the  given  values  of  y  are 
(m  —  2.4),  (3  m  —  7.6),  (10  vi  —25).  It  is  usual  to  assume  that  that  line 
for  which  the  su7n  of  the  squares  of  these  differences 

S=(m-  2.4)2  +  (3  m-  7.6)2  +  (lO  m  -  25)2 
is  least  is  the  best  compromise.     Show  that  this  would  give  m  =  2.50 
(nearly).    Draw  the  figure. 

19.  In  an  experiment  on  an  iron  rod  the  amount  of  stretching  s  (in 
thousandths  of  an  inch)  and  the  pull  p  (in  hundreds  of  pounds)  were 
found  to  be  (p  =  5,  s  =  4),  (p  =  10,  s  =  8),  {p  =  20,  s  r=  17).  Find  the 
best  compromise  value  for  m  in  the  equation  s  =  m  •  p,  under  the  assump- 
tion of  Ex.  18.  A71S.    About  5/6. 

20.  A  city's  bids  for  laying  cement  sidewalks  of  uniform  width  and 
specifications  are  as  follows:  Job  No.  1:  length  =  260  ft.,  cost,  $110; 
Job  No.  2:  length,  600  ft.,  cost,  §250;  Job  No.  3:  1500  ft.,  cost,  §630. 
Find  the  price  per  foot  for  such  walks,  under  the  assumption  of  Ex.  18. 
How  much  does  this  differ  from  the  arithmetic  average  of  the  price  per 
foot  in  the  three  separate  jobs  ? 

21.  The  amount  of  water  in  a  .standpipe  reaches  2000  gal.  in  250  sec, 
5000  gal.  in  610  sec.  From  this  information  (which  may  be  slightly 
faulty)  find  the  rate  at  which  water  was  flowing  into  the  tank,  under 
assumption  of  Ex.  18. 

22.  The  values  1  in.  =  2.5  cm.,  1  ft.  =  30.5  cm.  are  frequently  quoted, 
but  they  do  not  agree  precisely.  The  number  of  centimeters  c,  and  the 
number  of  inches  i,  in  a  given  length  are  surely  connected  by  an  equation 
of  the  form  c  =  ki.  Show  that  the  assumptions  of  Ex.  18  give  k  =  2.641. 
Is  this  the  same  as  the  average  of  the  values  in  the  two  cases  ?  Which 
result  is  more  accurate  ? 


70  APPLICATIONS   OF  DIFFERENTIATION      [IV,  §  41 

23.  In  experiments  on  the  velocity  of  sound,  it  was  found  that  sound 
travels  1  mi.  in  5  sec,  3  mi.  in  14.5  sec.  These  measurements  do  not 
agree  precisely.  Show  that  the  compromise  of  Ex.  18  gives  the  velocity 
of  sound  1084  ft.  per  second.  How  does  this  compare  with  the  average 
of  the  two  velocities  found  in  the  separate  experiments  ? 

24.  A  quantity  of  water  which  at  0°  C.  occupies  a  volume  vo,  at  6°  C. 
occupies  a  volume 

V  =  Uo(l  -  10-*  X  .5758  d  +  10-5  X  .756  ff^  -  lO"'  x  .351  6^). 

Show  that  the  volume  is  least  (density  greatest),  at  4°  C.  (nearly). 

25.  Determine  the  rectangle  of  greatest  perimeter  that  can  be  in- 
scribed in  a  given  circle.     Is  there  any  minimum  ? 

26.  What  is  the  largest  rectangle  that  can  be  inscribed  in  an  isosceles 
triangle  ?     Is  there  any  minimum  ? 

27.  Find  the  area  of  the  largest  rectangle  that  can  be  inscribed  in  a 
segment  of  the  parabola  ?/-  =  4  ax  cut  off  by  the  line  x  =  h. 

28.  Determine  the  cylinder  of  greatest  volume  that  can  be  inscribed 
in  a  given  sphere.     Is  there  also  a  minimum  ? 

29.  Determine  the  cylinder  of  greatest  convex  surface  that  can  be 
inscribed  in  a  sphere.     Is  there  a  minimum  ? 

30.  Determine  the  cylinder  of  greatest  total  surface  (including  the 
area  of  the  bases)  that  can  be  inscribed  in  a  given  sphere. 

31.  What  is  the  volume  of  the  largest  cone  that  can  be  inscribed  in  a 
given  sphere  ? 

32.  What  is  the  area  of  the  maximum  rectangle  that  can  be  inscribed 
in  the  ellipse  xVa-  +  y-Zb"^  =  1? 

PART   II.     RATES 

42.  Time-rates.  All  the  applications-  of  derivatives  are 
rates  of  increase  (or  decrease)  of  some  quantity  with  respect  to 
some  other  quantity  which  is  taken  as  the  standard  of  com- 
parison, or  independent  variable. 

Among  all  rates,  those  which  occur  most  frequently  are 
time-rates,  that  is,  rate  of  change  of  a  quantity  with  respect  to 
the  time. 


IV,  §  45]  RATES  71 

43.  Speed.  Thus  the  speed  of  a  moving  body  is  the  time- 
rate  of  increase  of  the  distance  it  has  traveled : 

(1)  V  =  sjyeed  *  =  lim  —  =  — , 
as  in  §  7,  p.  12,  and  in  numerous  examples. 

44.  Tangential  Acceleration.  The  specfZ  itself  may  change; 
the  time-rate  of  change  of  speed  is  called  the  acceleration  along 
the  path,  or  the  tangential  acceleration.f 

(2)  jr  =  tangential  acceleration  f  =  lim  — •  =  —  • 

A/  =  oA^       dt 


Thus  for  a 

body 

falling 

from  rest,  if  g  represents  the  gravitational 

constant, 

s=lgt^; 

hence 

^  =  1-^' 

and 

^^-|-  = 

it  follows  that  the  tangential  acceleration  of  a  body  falling  from  rest  is 
constant ;  that  constant  is  precisely  the  gravitational  constant  g.  J 

In  obtaining  the  tangential  acceleration,  we  actually  differ- 
entiate the  distance  s  twice,  once  to  get  v,  and  again  to  get 
dv/dt  or  _/„  hence  the  tangential  acceleration  is  also  said  to  be 
the  second  derivative  of  the  distance  s  passed  over. 

45.  Second  Derivatives,  Flexion.  It  often  happens,  as  in 
§  44,  that  we  wish  to  differentiate  a  function  twice.     In  any 

*  The  speed  v  is  distinguished  from  the  velocity  v  by  the  fact  that  the  speed 
does  not  depend  on  the  direction  ;  when  we  speak  of  velocity  we  shall  always 
denote  it  by  v  (in  black-faced  type)  and  we  shall  specify  the  direction. 

t  The  general  acceleration  J  is  also  a  directed  quantity  ;  when  we 
speak  of  the  acceleration  J  (not  tangential  acceleration  j  j)  we  shall  denote 
it  by  J,  and  give  its  direction.  As  in  the  case  of  speed,  the  letter  J,  iu  italic 
type,  denotes  the  value  oij  without  its  direction.     (See  Ex.  17,  p.  74.) 

t  The  value  of  c/  is  approximately  32.2  ft.  per  second  per  second  =  981  cm 
per  second  per  second. 


72  APPLICATIONS  OF  DIFFERENTIATION      [IV,  §  45 

case,  given  y  =  f(x),  the  slope  of  the  graph  is 

m=  -^=  lim  — ^« 

dx     AcB=o^aj 

The  slope  itself  may  change  (and  it  always  does  except  on  a 
straight  line)  ;  the  rate  of  change  of  the  slope  with  respect  to  x 
will  be  called  the  flexion  *  of  the  curve: 

0=  flexion  =■  —  =  hm , 

dx      Aa!=oAa; 

and  will  be  denoted  by  h,  the  initial  letter  of  the  word  hend. 

Thus  for  y  =  a;2,  m=2x,  &  =  2 1 ;  iox  y  =  oi?,  m  =  3  x^,  6  =  6  x  ;  for 
y  =  x^  —  12  X  +  7,  m  =  3  x2  —  12,  &  =  6  X ;  for  any  straight  line  y  =  kx+c, 
m  =  k,  6  =  0. 

The  value  of  b  is  obtained  by  differentiating  the  given  func- 
tion twice ;  the  result  is  called  a  second  derivative,  and  is 
represented  by  the  symbol : 

d^y  _  d_  fdy\  _  dm  _  , 


dJT      dx  \dxj      dx 
Likewise,  the  tangential  acceleration  in  a  motion  is 
d^s      d  /'ds\      dv 


df     dt\dt)      dt    '"^'" 
If  the  relation  between  s  and  t  is  represented  graphically, 
the  speed  is  represented  by  the  slojye,  the  tangential  acceleration 
by  the  flexion,  of  the  graph.     Thus  if  s  =  gt'/2  be  represented 
graphically,  as  in  Fig.  6,  p.  13,  the  slope  of  the  graph  is 

m  =  sloj)e  =  —  =  gt  =  speed  =  v, 

and  the  flexion  of  the  graph  is 

b  =  flexion  =  - —  =  — ^^  =  —  =  9  =  tangential  acceleration  =jr- 
at       at       az 

*  The  word  ciirvature  is  used  in  a  somewhat  different  sense.    See  §  97,  p.  1(59. 

t  The  flexion  for  this  parabola  is  constant ;  note  that  this  means  the  rate  of 
change  of  m  per  unit  increase  in  x,  not  per  unit  increase  in  length  along  the 
curve.    See  §61,  p.  106. 


IV,  §45]  SECOND   DERIVATIVES  73 


EXERCISES  XV.    SECOND  DERIVATIVES  —  ACCELERATION 

[In  addition  to  this  list,  the  second  derivatives  of  some  of  the  functions 
in  the  preceding  exercises  may  be  calculated.] 

1.  Calculate  the  first  and  second  derivatives  in  the  following  exercises. 
Interpret  these  exercises  geometrically,  and  also  as  problems  in  motion, 
with  s  and  t  in  place  of  y  and  x : 

(a)  y  =  a;2  +  3a;-4.  (A)  j/ =  Vx  +  >/^M^. 

(6)  y=-x2  +  3x-4.  (Z)   y  =(2  -  3x)2  (3  +  x). 

(c)  y  =  2 x2  -  X  -  15.  (,„)  y=(x  +  2)8  (x2  -  1). 

(d)  y  =  -  2  x2  -  X  -  15.  („)  y  =  vT+x  -^  Vl  -  x. 

(e)  2/  =  x2-|x-21.  (o)  2/  =  ax +  6. 

(/)  y  =  x8  —  3  x2  +  1.  (p)  y  =  c  (a  constant), 

(fir)  y  =  2x8-3x2-36x-20.  (q)  y  =  ax^  +  bx  +  c. 

(A)  y  =  X*  -  8  x2  +  2.  (r)  y  =  c  (x  -  a)». 

(0  y  =  x4-2x3  +  5x2  +  2.  (s)  y  =(x  -  a)»(x  -  6)» 

(j)  y=(l  +  x)-^(l-x).  (0    y  =  ^x-*. 

2.  Show  that  the  flexion  of  a  straight  line  is  everywhere  zero. 

3.  Show  that  if  the  distance  passed  over  by  a  body  is  proportional  to 
the  time  the  tangential  acceleration  is  zero.  What  is  the  speed  in  this 
case? 

4.  Show  that  the  flexion  of  the  curve  y  =  ax2  +  6x  +  c  is  everywhere 
the  same,  and  equal  to  twice  the  coefficient  of  x2. 

5.  Show  that  if  the  space-time  equation  is  s  =  at^  +  bt  -\-  c,  the  ac- 
celeration is  always  the  same  and  equal  to  twice  the  coefficient  of  t-.  Is 
such  a  motion  at  all  liable  to  occur  in  nature  ? 

6.  Find  the  flexion  of  the  curve  y  =  1/x.  Show  that  it  resembles  y 
itself  in  some  ways.  Does  the  slope  also  resemble  y  ?  "Which  one  re- 
sembles y  the  more  closely  ? 

7.  Can  you  interpret  Ex.  6  as  a  motion  problem  ?  "What  is  true  at  the 
beginning  of  the  motion  («  =  0)  ?  Can  a  curve  with  a  vertical  asymptote 
represent  a  motion  ?     Can  a  piece  of  such  a  curve  ? 

8.  Find  the  flexion  of  the  curve  y  =(x  -  2)8  (x  +  3)2  (x  -  4).  Show 
that  the  flexion  has  a  factor   (x— 2),   while  the  slope  has  a  factor 

(X  _  2)2  (X  +  3). 


74          APPLICATIONS   OF  DIFFERENTIATION      [IV,  §  45 

9.    Show  that  the  flexion  of  the  curve  y  =  (x  -  ay  (yfi  -ir  5)  has  a  fac- 
tor (x  -  a). 

10.  If  the  function  y  =  x^  +  ay:^  +  bx^  +  ex- +  dx  +  e  =  0  has  a  factor 
(a:  —  ay,  show  that  dy/dx  has  a  factor  (a;  —  «)2,  and  d-y/dx"^  has  a  factor 
(x-a). 

11.  If  the  equation  x^  +  ax*  +  fex^  +  cx^  +  rfx  +  e  =  0  has  a  triple  root 
X  =  «,  show  that  the  equation  20  x^  +  12  ax^  +  6  6.>:  +  2  c  =  0  has  a  factor 
X-  «. 

12.  Show  how  to  find  the  double  and  triple  roots  of  any  algebraic 
equation  by  the  Highest  Common  Divisor  process. 

13.  If  the  equations  of  the  curve  in  parameter  form  are  x  =  J^,  y  =  t^, 
find  the  slope  m  and  the  flexion  b  in  terms  of  t. 

[Hint.  First  find  m ;  then  use  the  values  of  7n  and  x  iu  terms  of  t  to  find 
d7n/dx.] 

14.  Find  m  and  b  for  each  of  the  following  parameter  forms : 

t^     [See  Ex.  13.] 

^         1  +  «3  ' 

15.  If  the  equations  of  Ex.  13  express  the  position  of  a  moving  par- 
ticle at  the  time  t,  find  the  horizontal  speed  Vx  —  dx/dt  and  the  vertical 
speed  Vy  —  dy/dt.  A  second  differentiation  gives  the  time-rates  of  change 
of  these  component  speeds  :  jx  =  dvx/dt  =  d'^x/dfi  and  jy  =  dvy/dt  = 
d-y/dfi.  Eind  each  of  these  quantities  in  Ex.  13.  In  each  of  the  exercises 
in  Ex.  14. 

16.  The  total  speed  v  =  Vt^-  +  Vy-  can  be  found  as  in  Ex.  7,  p.  49, 
from  the  values  of  v^  and  Vy.  Find  v  in  each  of  the  examples  of  Exs.  13 
and  14. 

17.  The  component  accelerations  jx  ancb-^'^  of  Ex.  15  may  be  com- 
bined to  get  the  total  acceleration  .;  ~  y/jx^  +  j/  by  the  so-called  paral- 
lelogram law  of  physics.    Find  j  in  each  of  the  examples  of  Exs.  13  and  14. 

18.  The  tangential  acceleration  jr  can  be  found  directly  from  Ex.  16, 
by  means  of  its  definition  jy=  dv/dt.  Find  j^  in  Exs.  13  and  14,  Show 
that  j  J,  and  j  are  different  in  every  exercise  except  14  (a). 

[The  reason  for  this  difference  is  not  difficult :  J^,  is  the  acceleration  in  the 
path  itself;  j  is  the  total  acceleration,  part  of   its  effect  being  precisely  to 


(a)  x-a  +  bt,y  =  c  +  dt. 

(b)  x  =  t\y^ 

(c)  z  =  t,y  =  r2  -,1-1  and  2. 

id)  X  =  1  +  «, 

«^=rii-4-'^'-- 

^■^>^  =  iT7. 

IV,  §  47]  CONCAVITY  75 

make  the  path  curved ;  hence  a  part  of  J  is  expended  not  to  increase  the  speed, 
but  to  change  the  direction  of  the  speed,  i.e.  to  bend  the  path.  Notice  that 
Ex.  14  (a)  represents  a  straight  line  path;  on  it  jj,=j;  this  holds  only  on 
straight  line  paths.     In  uniform  motion  on  a  circle,  for  example,  jj.=  0.] 

46.  Concavity.  Points  of  Inflexion.  If  the  flexion  b  = 
dm/dx  is  positive,  the  slope  is  increasing,  and  the  curve  turns 
upwards,  or  is  concave  iipivards;  if  the  flexion  is  negative,  the 
slope  is  decreasing,  and  the  curve  is  concave  doicmcards. 

Thus  y  =  z'^  is  concave  upwards  everywhere,  since  b  =  2  is  positive. 
For  y  =  x^  we  find  b  =  G  .r,  which  is  positive  when  x  is  positive,  and  nega- 
tive when  X  is  negative  ;  hence  y  —  x^  is  concave  upwards  at  the  right, 
and  concave  downwards  at  the  left  of  the  origin. 

A  point  at  which  the  curve  changes  from  being  concave  up- 
wards to  being  concave  downwards,  or  conversely,  is  called  a 
point  of  inflexion. 

The  value  of  the  flexion  h  changes  from  positive  to  negative, 
or  conversely,  in  passing  such  a  point ;  hence  the  value  of  h  at 
(I  point  of  injlexion  is  zero,  if  it  has  any  value  there.* 

Thus  the  origin  is  a  point  of  inflexion  on  the  curve  y  =  x^,  for  the 
curve  is  concave  downwards  on  the  left,  concave  upwards  on  the  right,  of 
the  origin. 

47.  Second  Test  for  Extremes.  In  seeking  the  extreme 
values  of  a  function  y  =f(x),  we  find  first  the  critical  j)oints 
(§  38,  p.  63),  i.e.  the  points  at  which  the  tangent  is  horizontal. 

If,  at  a  critical  point,  &  =  d-y/dx''  >  0,  the  curve  is  also  con- 
cave upwards,^  and  the  function  has  a  minimum  there;  if  &<0, 
the  curve  is  concave  doicnwards,  and  f(x)  has  a  maximuvi; 
that  is, 

•/•  dif     ri        7  1       d-ri(>0]    ,  r/  \  •         (minimuin  ] 

if  m=  -^=0  and  6  =  — ^  [at  x=a,  f(a)  is  a  i  .  \. 

dx  dx-[<()j  [maximum) 

*  Points  where  the  tangent  is  vertical,  for  example,  may  be  points  of  in- 
flexion. 

t  The  curve  is  then  also  concave  upwards  on  both  sides  of  the  point ;  if  the 
curve  is  concave  upwards  on  one  side  and  downwards  on  the  other,  b  must  be 
zero  if  it  exists  at  the  point. 


76  APPLICATIONS  OF  DIFFERENTIATION      [IV,  §  48 

Whenever  the  flexion  is  not  zero  at  a  critical  point,  this 
method  usually  furnishes  an  easy  final  test  for  extremes.  If 
the  flexion  is  zero,  no  conclusion  can  be  drawn  directly  by  this 
method.*     (See,  however,  §  135.) 

48.  Illustrative  Examples. 

Example  1.  Consider  the  function  y  =  x^  —  12  x  +  7.  See  Ex.  3, 
p.  10,  and  Ex.  1,  p.  65.     The  slope  and  the  flexion  are,  respectively, 

dx  dx-      dx 

The  critical  points  are  (see  Ex.  1,  p.  65)  x  =  ±  2.  Since  6  a;  is  positive 
when  X  is  positive,  b  is  positive  for  x  >  0 ;  likewise  6  <  0  when  x  <  0. 
Hence  the  curve  is  concave  upwards  when  x  >  0,  and  concave  downwards 
when  X  <  0.  At  x  =  +  2,  &  >  0,  hence  by  §  47,  y  has  a  minimum  at  x  = 
+  2;  at  x=— 2,  &<0,  hence  y  has  a  maximum  (compare  p.  10  and 
p.  66). 

To  find  a  point  of  inflexion  first  set  6  =  0  ; 

,      dm      d-y      -j         a     •  « 

6  =  —  =  — ^  =  6  X  =  0,    I.e.  X  =  0. 
dx       dx^ 

Since  dm/dx  is  negative  for  x  <  0  and  positive  for  x  >  0,  the  given  curve 
is  concave  downwards  on  the  left  and  concave  upwards  on  the  right  of 
this  point ;  hence  x  =  0,  j/  =  7  is  a  point  of  inflexion.  (See  Fig.  17, 
and  §  49,  p.  77.) 

Example  2.     Consider  the  function  y=Sx*-12  x^+50  (Ex.  2,  p.  65). 
The  slope  and  the  flexion  are,  respectively, 

,n  =  ^  =  12x3  ^  36x2  ;     b  =  ^  =  '^  =  36x2-  72  x. 
dx  dx      dx^ 

The  critical  points  are  x  =  0,  x  =  3.     At  x^  3,  6  =  108  >  0,  hence  y  is 

a  minhnum  there.     At  x  =  0,  6  =  0,  and  no  conclusion  is  reached  by  this 

method  (compare,  however,  p.  65).     To  find  points  of  inflexion,  first  set 

6  =  0; 

6  =  —  =  ^  =  36  x2  -  72  X  =  0,  i.e.  x  =  0  or  x  =  2. 
dx       dx'^ 

*  Even  in  this  case  one  may  decide  by  determining  whether  the  curve  is 
concave  upwards  or  downwards  ou  both  sides  of  the  point ;  but  the  method  of 
§  40  is  usually  superior. 


IV,  §  49] 


DERIVED  CURVES 


77 


Near  x  =  0,  at  the  left,  dm/dx  =  36x(x  —  2)  is  positive,  at  the  right,  nega- 
tive ;  the  given  curve  is  concave  upwards  on  the  left,  downwards  on  the 
right,  and  (x  =  2,  j/  =  2)  is  a  point  of  inflexion.     (See  Fig.  15,  and  §  49.) 

Example  3.     For   a   body   thrown  vertically  upwards,  the  distance  s 
from  the  earth  is  : 

s  =  -l^gfi  +  vot, 

where  Vo  is  the  speed  with  which  it  is  thrown. 

The  speed  and  the  tangential  acceleration  are,  respectively, 


=  -gt 


dt^' 


g- 


If  we  draw  a  graph  of  the  values  of  s  and  t,  the  speed  v  (slope  of  the 

graph)  is  zero  when 

»=—</«  +  Vo  =  0,     i.e.  t  =  vo/g., 

that  is,  the  point  is  a  critical  point  on  the  graph.  The  tangential  acceler- 
ation (flexion  of  the  graph)  is  negative  everywhere,  hence  the  graph  is 
concave  dowmoards. 

In  particular  at  the  critical  point  just  found,  b  is  negative  ;  hence  « 
has  a  maximum  there  : 


.=-i.. 


Vot 


1^'',    wheni 
^  9 


The  figure,  for  the  special  values  Vq  —  (34  and  g  —  32,  is  drawn  in  §  49. 


49.   Derived  Curves.     It  is  very  instructive 
same  figure  graphs  which  give  the 
values  of  the  original  function,  its 
derivative,  and  its  second  deriva- 
tive. 

These  graphs  of  the  derivatives 
are  called  the  derived  curves  ;  they 
represent  the  .s/ope  (or  i^peed  in  case 
of  a  motion)  and  the  flexion  (or 
tangential  acceleration). 

The  figures  for  the  curves  of  Exs.  1 
and  2  of  §  48  are  appended.  The  stu- 
dent should  show  that  each  statement 
made  in  §  48  and  each  statement  made 


to  draw  in  the 


Fig 


78         APPLICATIONS  OF  DIFFERENTIATION     [IV,  §  49 


V                              y<^      ^s 

\                    ^ 

s 

50      ^^          ^ 

s 

^     ^  /             : 

\ 

\^ 

t\ 

:      "s 

1     ^v 

\ 

25     t    ^^in 

^ 

1        '\ 

^ 

t            \Z- 

i^ 

,             \ 

T 

t             ^^ 

\ 

t               "v     : 

0/                     1                     2  \ 

o              -iT 

/                    S:-.       -     Wt^      aT^I^K 

V 

/            i'-=  -321     H^      ^^ 

"^ 

/              /^=-32                          ^ 

Si" 

~9S                       jl 

'^               ^ 

.20--          --^-                   - 

^           ^ 

-f~ 

-      \                v 

IV,  §49]  DERIVED   CURVES  79 

on  p.  65,  for  each  of  the  examples,  is  iUustrated  and  verified  in  these 
figures. 

The  similar  curves  for  space,  speed,  and  acceleration  are  drawn  in 
Fig.  19,  for  the  motion  of  a  body  thrown  upwards  : 

s  =—l  gt2  +  vot  for  g  =  32,  vq  =  64. 

Verify  the  statements  made  in  Ex.  3,  §  48. 

In  drawing  such  curves,  the  second  derivative  should  be 
drawn  first  of  all ;  the  information  it  gives  should  be  used  in 
drawing  the  graph  of  the  first  derivative,  which  in  turn  should 
be  used  iu  drawing  the  graph  of  the  original  function. 

EXERCISES  XVI.  —  FLEXION  —  DERIVED   CURVES 

1.  Draw,  in  the  order  just  indicated,  the  first  and  second  derived 
curves  in  Ex.  1  (a).  List  XIV,  p.  67  ;  and  show  that  each  step  of  your 
work  in  that  example  Is  exhibited  by  these  figures. 

2.  Draw  the  derived  curves  for  Exs.  1  (6),  1  {d),  1  (/),  1  (?i)  of 
List  XIV,  p.  67  ;  and  show  their  connection  with  your  previous  work. 

3.  Draw  the  original  and  the  derived  curves  for  the  function 
y  =  x^  —  'd  x^  +  \bx  —  Q.  Find  the  extreme  values  of  y,  and  explain  the 
figures.  For  what  value  of  x  is  the  flexion  zero  ?  Does  this  give  a  point 
of  inflexion  on  the  original  curve  ? 

'  4.    Find  the  extreme  values  of  y  and  the  points  of  inflexion  on  the 
following  curves  ;  in  each  case  draw  complete  figures  : 

(«)  2/  =  2  x5-  3x-^  -  36  X.  (/)  y  =  Ax^  +  Bx  +  C. 

(fe)  y  =  ix^  —  X-  —  24  X.  (g)    y  =  mx  +  n. 

(c)   y  =  x^  +  x2.  (/i)    y  =  yJx. 

.  (d)  yz=x*-2  x^  +  40.  (0     2/  =  •'''  +  px  +  q. 

(e)   y  =  x(x  +  2y.  (j)    y  .=  x^  +  W/x. 

5.  Show  that  the  flexion  of  the  hyperbola  x>j  —  a-  varies  inversely  as 
the  cube  of  the  abscissa  x. 

6.  Show  that  the  flexion  of  the  conic  Ax'^  +  By-  =  1  (ellipse  or 
hyperbola)  varies  inversely  as  the  cube  of  the  ordinate  y. 

7.  What  is  the  effect  upon  the  flexion  of  changing  the  sign  of  a  in  the 
equation  y  =  ax^  +  bx  +  c? 

8.  A  beam  of  uniform  depth  is  said  to  be  of  "uniform  strenuth  "  (in 
resisting  a  given  load)  if  the  actual  shape  of  its  upper  surface  under  the 
load  is  of  the  form  y  =  ax-  +  bx  +  c,  where  x  and  y  represent  horizontal 


80  APPLICATIONS  OF  DIFFERENTIATION      [IV,  §  49 

and  vertical  distances  measured  from  the  middle  point  of  the  beam's 
surface  in  its  original  (unbent)  position.  Show  that  the  flexion  of  such 
a  beam  is  constant. 

9.    Show  that  the  addition  of  a  constant  to  the  value  of  y  does  not 
affect  the  slope  nor  the  flexion. 

10.  Show  that  the  addition  of  a  term  of  the  form  kx  +  c  to  the  value 
of  y  does  not  affect  the  flexion.     What  effect  does  it  have  upon  the  slope  ? 

11.  Show,  by  means  of  Exs.  9  and  10,  that  any  beam  in  which  the 
flexion  is  constant  has  the  form  specified  in  Ex.  8. 

12.  Show,  by  a  process  precisely  similar  to  that  of  Ex.  11,  that  a 
motion  in  which  the  tangential  acceleration  is  constant  is  defined  by  an 
equation  of  the  form  s  =  at-  -\- ht  +  c. 

13.  What  is  the  effect  upon  the  graph  of  an  equation  if  a  constant  is 
added  to  ?/  ?  How  are  the  positions  of  the  maxima  and  minima  affected  ? 
[Take  into  account  vertical  as  well  as  horizontal  displacement.] 

14.  What  is  the  effect  upon  the  points  of  inflexion  if  a  term  kx  -\-  c'ls 
added  to  the  value  oi  y '>  Will  this  change  in  the  original  curve  change 
the  values  of  x  which  correspond  to  extreme  values  of  y  ? 

15.  Show  that  the  curve  (1 +  x-)2/ =  (1  —  a;)  has  three  points  of 
inflexion  which  lie  on  a  straight  line. 

16.  Show  that  the  graph  of  a  polynomial  of  the  ?i'i>  degree  cannot  have 
more  than  n—2  points  of  inflexion. 

17.  Show  that  if  a  polynomial  has  a  factor  (a;  — a)*,  its  flexion  has  a 
factor  (x  -  a)*-2. 

18.  Find,  by  the  methods  of  Exs.  9-12,  what  the  form  of  y  must  be  if 
the  slope  is: 

(a)^  =  0;     (6)^/  =  _3;     {c)^  =  Qx;    (d)  '^  =  ax  +  b. 
^  ^  dx  ^  ^  dx  '     ^  ^  dx  '    ^  ^  dx 

19.  What  is  the  form  of  y  if  the  flexion  i^  6  ?  if  the  flexion  is  2  x  +  3  ?   y 
if  the  flexion  is  zero  ? 

20.  If  a  beam  of  length  I  is  supported  only  at  both  ends,  and  loaded 
by  a  weight  at  its  middle  point,  its  deflection  y  at  a  distance  x  from  one 
end  is  y  =  k  (3Z'^x  —  4  x^),  provided  the  cross  section  of  the  beam  is  con- 
stant. Find  the  flexion  and  show  that  there  are  no  points  of  inflexion 
between  the  supports. 

21.  If  the  beam  of  Ex.  20  is  rigidly  fixed  at  both  ends,  and  loaded  at 
its  middle  point,  the  deflection  of  each  half  of  the  beam  is  y  =  k  (SZx^— 4x^), 
where  x  is  measured  from  either  end.     Show  that  there  is  a  point  of 


IV,  §  51]  TIME-RATES  81 

inflexion  at  a  distance  Z/4  from  the  end,  and  that  the  greatest  deflection 
is  at  the  middle  point. 

22.  Find  the  points  of  inflexion  and  the  point  of  maximum  deflection 
of  a  uniform  beam  of  length  I  whose  deflection  is : 

(a)  y  =  k(_3lx^-x^). 

[Beam  rigidly  embedded  at  one  end,  loaded  at  other  end.     Origin  at  fixed 
end.] 

(6)  y  =  k(Sx-l^-2xi). 

[Beam  freely  supported  at  both  ends,  loaded  uniformly.      Origin   at 
lowest  point.] 

(c)  2/  =  ^-  (6  Z%-^  -ilx^-\-x*). 

[Beam  embedded  at  one  end  only  ;   loaded  uniformly.     Origin  at  fixed 
end.] 

(d)  y=k  {fx  -  3  ix3  +  2  a:*). 

[Beam  embedded  at  one  end,  supported  at  the  other  end ;  loaded  uni- 
formly.    Origin  at  free  end.] 

50.  Angular  Speed.  If  a  wheel  tvirns,  a  given  spoke  of  it 
makes  an  angle  6  with  its  original  position  which  changes  with 
the  time,  i.e.  ^  is  a  function  of  the  time : 

d  =  f{t). 
Tlie  time-rate  of  change  of  the  angle  6  is  called  the  angular 
speed ;  it  is  denoted  b>j  w : 

o)  =  anqular  speed  =  ^=  lini  — . 

51.  Angular  Acceleration.  The  angular  speed  may  change; 
the  time-rate  of  change  of  the  angular  speed  is  called  the  angular 
acceleration ;  it  is  denoted  by  a : 

^ .     Aw  _  do)  _  d^O 
a  =  angular  acceleration  =  am  T7  "~  ^ "~  '^ ' 

Example  1.  A  flywheel  of  an  engine  starts  from  rest,  and  moves  for 
30  seconds  according  to  the  law 

18U(J         30 
■where  e  is  measured  in  degrees,  after  which  it  rotates  uniformly. 


82 


APPLICATIONS  OF  DIFFERENTIATION     [IV,  §  51 


Then 
and 


de 


dt 

doj 
dt 


—  <3  +  l(2 

450  10 


150 


This  example  furnishes  an  instance  in  which  the  derived  curves,  i.e., 
the  graphs  which  show  the  values  of  w  and  of  a  are  more  important  than 

the  original  curve;  for  the 
total  angle  described  is  rela- 
tively unimportant. 

In  the  figure  a  scale    is 
chosen  which  shows  partic- 
ularly well  the  variation  of 
w  ;  ^  is  allowed  to  run  off  of 
the   figure  completely,  since 
its  values  are  uninteresting. 
The  acceleration  a  is  so 
arranged    that    it  does  not 
suddenly  drop  to  zero  when 
the  flywheel  is  allowed  to  run 
uniformly  ;   and    the   values 
of  «  are  never  large.     Some- 
thing resembling  this  figure 
is  what  actually   occurs   in 
starting  a  large  flywheel. 
In  actual  practice  with  various  machines,  curves  of  this  type  are  often 
drawn  experimentally ;  the  equations  serve  only  as  approximations  to  the 
reality ;   but   they   are  often  indispensable  in  calculating   other  related 
quantities,  such  as  the  acceleration  in  this  example. 

Curves  which  resemble  the  graph  of  w  in  this  example  occur  frequently. 
(See  §§  87,  134.)  , 

52.  Momentum.  Force.  As  a  further  illustration  of  time- 
rates,  we  mention  a  statement  often  given  as  the  definition  of 
force:  force  is  the  time-rate  of  change  of  momentum.  (Compare 
Newton's  Second  Law  of  Motion.) 

The  momentum  3f  of  a  body  moving  in  a  straight  path  is 
defined  as  the  product  of  the  mass  m  of  the  body  times  its 
speed  v.  M  =m  ■  v. 


\[    \ 

A^le        1 

(degrees)     | 

t 

-0                      t 

03 

.0                   -|^ 

^        ''s 

IT 

J^         ^ 

t          t 

V                       ^ 

7 

W 

2n                T               Z 

T      7  ' 

9i|lo5«*  4-    —^f' 

t     : 

y 

^  =  ^-i|!  --  iV'' 

IT        ^ 

10               ^      ^ 

x  =  -4-i!j-  \i 

h~l 

1/^ 

.  2^^-    '^- 

-^0                    10                  2 

30^^            40 

Fig.  20. 


IV,  §  52]  TIME-RATES  83 

The  force  acting  on  the  body  is  therefore 

^     clM      V      A  3/      d(m-v)  dv  .  (Ps 

dt       At^o  ^t  dt  dt  "^  de 

This  law  is  often  stated  in  the  form  :  the  force  is  the  product 
of  the  mass  times  the  acceleration;  for  the  present  the  results 
are  stated  only  for  a  body  moving  in  a  straight  line  along 
which  the  force  itself  acts. 

This  consideration  of  time-rates  makes  clear  that  the  two 
delinitions  of  force  quoted  above  are  equivalent. 

EXERCISES   XVn.— TIME-RATES 

1.  Express  as  a  time-rate  the  speed  d  of  a  moving  body,  and  write  the 
result  as  a  derivative. 

2.  Express  as  time-rates  the  following  concepts  : 

(a)  The  tangential  acceleration  of  a  moving  body. 

(b)  The  horizontal  speed  of  a  moving  body. 

(c)  The  vertical  speed  of  a  moving  body. 

((i)  The  speed  of  evaporation  of  a  liquid  exposed  to  air. 
(e)  The  speed  of  formation  of  rust  on  iron. 
(/)  The  rate  of  growth  of  the  height  of  a  tree. 
(g)  The  rate  of  fluctuation  of  the  value  of  gold, 
(/t)  The  rate  of  rise  or  fall  of  the  height  of  a  river. 

3.  In  Ex.  2,  which  of  the  rates  mentioned  are  surely  constant ;  which 
may  possibly  be  constant  in  some  instances  ;  which  may  be  constant  part 
of  the  time  ?  For  which  of  them  does  a  concept  analogous  to  acceleration 
have  a  meaning  ? 

4.  If  such  a  rate  is  constant,  how  can  the  total  amount  (or  value)  of 
the  changing  quantity  be  computed  ?  Find  the  total  amount  of  water  in 
a  tank  which  originally  contained  2000  gal.,  after  water  has  run  into  it 
for  10  min.  at  the  rate  of  10  gal.  a  second. 

5.  If  a  train,  after  it  is  10  mi.  from  Chicago,  travels  directly  away 
at  60  mi.  an  hour,  how  far  is  the  train  from  Chicago  5  min.  later  ? 

6.  If  y  is  any  varying  quantity,  and  if  dy/dt  =  7,  express  y  in  terms 
of «  if  y  =  10  when  t  =  0.     Again,  if  ?/  =  5  when  t  =  0. 

7.  If  dy/dt  =  2t  +  S,  express  y  in  terms  ottiiy  =  0  when  t  =  0.  [See 
Exs.  9,  10,  List  XVI.] 


84  APPLICATIONS  OF   DIFFERENTIATION     [IV,  §  52 

8.  A  flywheel  rotates  so  that  0  =  t^  ~  1000,  where  0  is  the  angle  of 
rotation  (in  degrees)  and  t  is  the  time  (in  seconds).  Calculate  the 
angular  speed  and  acceleration,  and  draw  a  figure  to  represent  each  of 
them. 

9.  Suppose  that  a  wheel  rotates  so  that  d  =  t^  ^  1000  where  6  is 
measured  in  radians  [1  radian  =  180°/t].  Is  its  speed  greater  than  or 
less  than  that  of  the  wheel  in  Ex.  8  ?  What  is  the  ratio  of  the  speeds 
in  the  two  cases  ? 

10.  Compare  linear  speeds  in  miles  per  hour  with  speeds  in  feet  per 
minute.     Reduce  60  miles  per  hour  to  feet  per  second. 

11.  Compare  angular  speeds  in  radians  per  second  to  speeds  in  degrees 
per  second.     Reduce  90°  per  second  to  radians  per  second. 

12.  Compare  angular  speeds  in  revolutions  per  minute  (U.  P.  M.)  with 
speeds  in  degrees  per  second.  Express  the  angular  speed  in  Example  1, 
§51,  in  R.  P.M. 

13.  Reduce  a  linear  acceleration  60  in./sec./sec.  to  ft./sec./sec. ;  to 
in./min./min.  ;  to  ft./min./min.  Express  the  acceleration  due  to  gravity 
(g  =  32.2  ft./sec./sec.)  in  each  of  these  units. 

14.  Reduce  the  angular  acceleration  in  Example  1,  §  51,  to  rev. /sec./ 
sec.  ;  to  rev./min./min. 

15.  If  a  wheel  moves  so  that  6  =  —  t*/16  —  i/32,  where  6  is  measured 
in  radians  and  t  in  minutes,  find  the  angular  speed  and  acceleration  in 
terms  of  radians  and  minutes  ;  in  terms  of  revolutions  and  minutes ;  in 
terms  of  radians  and  seconds  (of  time). 

16.  If  a  Ferris  wheel  turns  so  that  ^  =  20  f2  while  changing  from  rest 
to  full  speed,  where  6  is  in  degrees  and  t  in  minutes,  when  will  the  speed 
reach  20  revolutions  per  hour  ? 

17.  If  the  angular  speed  is  w  =  kt  as  in  Ex.  16,  show  that  the  accelera- 
tion a  is  constant.  Conversely,  show  that  if  a  =  k,  and  if  t  is  the  time 
since  starting,  u  =  kt.  * 

18.  How  far  does  a  point  on  the  rim  of  a  wheel  travel  during  one  com- 
plete revolution  ?  Express  the  linear  speed  of  a  point  on  the  rim  of  a 
wheel  10  ft.  in  diameter  when  the  angular  speed  is  4  R.  P.  M. 

19.  Express  in  miles  per  hour  the  speed  of  a  point  of  a  wheel  2  ft.  in 
diameter  which  is  rotating  with  an  angular  speed  of  10  revolutions  per 
second. 

20.  If  the  Ferris  wheel  of  Ex.  16  is  100  ft.  in  diameter,  what  is  the 
linear  speed  of  the  rim  at  20  R.  per  hour  ? 


IV,  §  53]  RELATED   RATES  85 

21.  Find  the  linear  speed  and  the  tangential  acceleration  of  a  poiiil  on 
the  rim  of  the  wheel  of  Ex.  1,  §  51,  if  the  wiieel  is  lU  ft.  in  diameter. 
What  are  they  when  «  =  30  sec.  ? 

22.  Find  the  linear  speed  and  acceleration  in  Ex.  8,  if  the  radius  of 
tlie  wheel  is  4  ft.  How  large  would  the  wheel  of  Ex.  9  have  to  be  to 
make  the  linear  speed  of  its  rim  the  same  ? 

23.  An  engine  with  driving  wheels  5  ft.  in  diameter  is  traveling  40 
mi./hr.    Express  the  angular  speed  of  the  rim  in  revolution  per  minute. 

24.  If  a  train  starts  from  a  station  with  speed  v  =  t/2  +  t'^/lOO  (in 
feet  and  seconds),  find  the  angular  speed  and  hence  the  angular  accelera- 
tion of  drivers  6  ft.  in  diameter.  What  is  the  value  of  each  of  these 
quantities  when  t  =  10? 

25.  Find  the  momentum  (  =  mass  x  speed)  of  a  falling  body,  if  the 
distance  passed  over  is  s  =  gt''/2.     Find  tlie  force  acting. 

[Note.  If  force  is  measured  in  pounds,  mass  =  weight  in  pounds 
-T-  g.     Hence,  force  =  mass  •  g.'] 

26.  If  a  body  moves  so  that  s  =  Zt-  —  12,  find  the  force  acting  if  the 
body  weighs  10  lb. 

27.  The  hammer  of  a  pile  driver  weighs  1000  lb.  If  it  drops  15  ft. 
onto  a  pile  according  to  the  law  of  Ex.  25,  what  is  the  momentum  of  its 
impact?  The  average  force  of  the  blow  is  the  average  rate  at  whicli  the 
momentum  is  destroyed.  How  much  is  this  if  the  hammer  is  stopped  in 
1/1000  sec.  ? 

28.  What  is  the  average  force  of  a  hammer  blow  by  a  2-lb.  hammer 
moving  at  30  ft./sec,  stopped  in  1/1000  sec.  ? 

29.  The  kinetic  energy  of  a  moving  body  \s  E  =  mv^/2.  Show  that 
ilE/dt  =  mv  ■  dv/dt  =  momentum  x  acceleration. 

30.  An  electric  current  c  (measured  in  amperes)  is  the  (luantity  q  of 
electricity  (in  coulombs)  which  passes  a  given  point  per  second.  Express 
this  fact  in  the  derivative  notation. 

53.  Related  Rates.  If  a  relation  between  two  quantitie.s  is 
known,  the  time-rate  of  change  of  one  of  them  can  be  expressed 
in  terms  of  the  time-rate  of  change  of  the  other. 

Thus,  in  a  spreading  circular  wave  caused  by  throwing  a 
stone  into  a  still  pond,  the  circumference  of  the  wave  is 

(1)  c  =  2  7rr, 


86  APPLICATIONS  OF  DIFFERENTIATION     [IV,  §  53 

where  r  is  the  radius  of  the  circle.     Hence 

(2)  '-^  =  2,r^; 

or,  the  time-rate  at  which  the  circumference  is  increasing  is 
2  7r  times  the  time-rate  at  which  the  radius  is  increasing. 
(Compare  Ex.  8,  p.  27.)     Dividing  both  sides  by  dr/dt,  we  find 

dc     dr     o         ^c       7       ^ 

= =  2-77  =  —  =dc^  dr\ 

dt      dt  dr 

that  is,  the  ratio  of  the  time-rates  is  the  derivative  of  c  with  re- 
spect to  r;  or,  the  ratio  of  the  time-rates  is  equal  to  the  ratio  of 
the  differentials. 

The  fact  just  mentioned  is  true  in  general ;  if  y  and  x  are 
any  two  related  variables  which  change  with  the  time,  it  is 
true  (Rule  [VII „],  P-  40)  that: 

^^'I^  =  'M  =  dy^dx, 
dt      dt      dx 

that  is,  the  ratio  of  the  time-rates  of  y  and  x  is  equal  to  the  ratio 
of  their  differentials,  i.e.  to  the  derivative  dy/dx. 

Example  1.  "Water  is  flowing  into  a  cylindrical  tank.  Compare  the 
rates  of  increase  of  the  total  volume  and  the  increase  in  height  of  the 
water  in  the  tank,  if  the  radius  of  the  base  of  the  tank  is  10  ft.  Hence 
find  the  rate  of  inflow  which  causes  a  rise  of  2  in.  per  second ;  and  find 
the  increase  in  height  due  to  an  inflow  of  10  cu.  ft.  per  second.  Consider 
the  same  problem  for  a  conical  tank. 

(u4)   The  volume  Fis  given  in  terms  of  the  height  h  by  the  formula : 

V  =  irr"-h  =  im  Tvh, 

hence  ^=100:r^; 

dt  dt 

or,  the  rate  of  increase  in  volume  (in  cubic  feet  per  second)  is  100  ir  times 
the  rate  of  increase  in  height  (in  feet  per  second). 

If  dh/dt  =  1/6  (measured  in  feet  per  second),  dv/dt  =  100  7r/6  = 
(roughly)  52.3  (cubic  feet  per  second).  If  dv/dt  =  10,  dh/dt  =  10-- 100  tt 
=  (roughly)  .031  (in  feet  per  second)  =  22.3  (in  inches  per  minute). 


IV,  §  53] 


RELATED  RATES 


87 


(B)   If  the  reservoir  is  coru'cal,  we  have 

V=  J  irr-/i  =  ^7r;inan2a, 

where  r  is  the  radius  of  the  water  surface,  h  the  height  of  the  water, 
and  a  the  half-angle  of  the  cone  ;  for  ?•  =  h  tan  «. 
this  case 


In 


=  irh'^  tan^  a 


which  varies  with  h.  If  «  =  45°  (tan  a  =  1),  at  a 
height  of  10  ft.,  a  rise  1/6  (feet  per  second)  would 
mean  an  inflow  of  ttA^x  (1/6)=  100  7r/6  =52.3  (cubic 
feet  per  second).  At  a  height  of  15  feet,  a  rise  of 
1/6  (feet  per  second)  would  mean  an  inflow  of  225  tt/6 
=  (roughly)  117.8  (cubic  feet  per  second).  An  inflow 
of  100  (cubic  feet  per  second)  means  a  rise  in  height 
of  lOO/irh-,  which  varies  with  the  height ;  at  a  height 
of  5  ft.,  the  rate  of  rise  is  i/w  =  1.28  (feet/second). 


Fig.  21. 
the  air  resistance, 


Example  2.     A  body  thrown  upward  at  an  angle  of 
45"^,  with  an  initial  speed  of  100  ft.  per  second,  neglecting 
etc. ,  travels  in  the  parabolic  path 

10000 
where  x  and  y  mean  the  horizontal  and  vertical  distances  from  the  start- 
ing point,  respectively;   g  is  the  gravitational  constant  =  32.2  (about); 
and  the  horizontal  speed  has  the  constant  value  100/ ■\/2.     Find  the  ver- 
tical speed  at  any  time  t,  and  find  a  point  where  it  is  zero. 

The  horizontal  speed  and  the  vertical  speed,  i.^.  the  time-rate  of  change 
of  X  and  y,  respectively,  are  connected  by  the  relation  (see  §§  8,  29.) 
dy  .  dx  _dy q^ 


dt 


hence 


dt 
gx 


dx 

dx 


dt      ^      5000  (It 

This  vertical  speed  is  zero  where 

50  V2       V2 


5000 
gx 


+  1; 


^100 
50  V2       V2 


5000 


155.3  (about), 


which  corresponds  to  y  =  2500/gf  =  77.7  (about).  At  this  point  the  verti- 
cal speed  is  zero  ;  just  before  this  it  is  positive,  just  afterwards  it  is  nega- 
tive. When  x  =  0  the  value  of  dy/dt  is  100/ \/2;  when  a:  =  2500/fir, 
dy/dt  =  50/\/2  ;  when  x  =  7500/gr,  dy/dt  =-  50/\/2. 


88  APPLICATIONS   OF   DIFFERENTIATION      [IV,  §  53 

EXERCISES   XVIII.  —  RELATED   RATES 

1.  Water  is  flowing  into  a  tank  of  cylindrical  shape  at  tlie  rate  of 
50  gal.  per  minute.  If  the  tank  is  8  ft.  in  diameter,  find  the  rate  of  in- 
crease in  the  height  of  the  water  in  the  tank. 

2.  Water  is  flowing  into  a  cone-shaped  tank,  20  ft.  across  at  the 
bottom  and  15  ft.  high,  at  the  rate  of  100  cu.  ft.  per  minute.  Calculate 
the  rate  of  increase  of  the  water  level. 

How  fast  is  the  water  entering  the  same  tank  when  the  height  is  G  ft., 
if  the  level  is  rising  6  in.  per  minute  ? 

3.  A  funnel  8  in.  across  the  top  and  6  in.  deep  is  being  emptied  at 
the  rate  of  2  cu.  in.  per  minute.  How  fast  does  the  surface  of  the  liquid 
fall  ? 

4.  A  hemispherical  bowl  1  ft.  in  diameter  and  full  of  water  is  being 
emptied  through  a  hole  in  the  bottom  at  the  rate  of  10  cu.  in.  per  second. 
How  fast  is  the  surface  of  the  water  sinking  when  100  cu.  in.  have  run 
out  ?     When  the  bowl  is  just  half  full  ? 

5.  If  water  flows  from  a  hole  in  the  bottom  of  a  cylindrical  can  of 
radius  r  into  another  can  of  radius  r',  compare  the  vertical  rates  of  rise 
and  fall  of  the  two  water  surfaces. 

6.  If  a  funnel  is  8  in.  wide  and  6  in.  deep  and  liquid  flows  from  it  at 
the  rate  of  5  cu.  in.  per  minute,  determine  the  time-rate  of  fall  of  the 
surface  of  the  liquid. 

^  7.  Compare  the  vertical  rates  of  the  two  liquid  surfaces  when  water 
drains  from  a  conical  funnel  into  a  cylindrical  bottle.  Compare  the  time- 
rate  of  flow  from  the  funnel  with  the  time-rate  of  the  decrease  of  the  wet 
perimeter. 

8.  If  a  wheel  of  radius  B  is  turned  by  rolling  contact  with  another 
wheel  of  radius  B',  compare  their  angular  speeds  and  accelerations. 

9.  If  a  gear  wheel  moves  a  toothed  pck  so  that  a  point  of  the  rack 
moves  according  to  the  equation  s  =  1  —  t/2  +  fi/3,  what  is  the  angular* 
velocity  and  angular  acceleration  of  the  wheel  at  any  time  t,  expressed  in 
revolutions  and  seconds?  Express  the  angular  speed  and  the  angular 
acceleration  in  terms  of  radians  and  seconds. 

10.  Compare  the  speed  of  a  train  with  tlie  speed  of  a  point  on  the  rim 
of  a  wheel  ;  compare  their  accelerations. 

11.  If  a  point  moves  on  a  circle  so  that  the  arc  described  in  time  t  is 
s  =  t'^  —  1/t-  +  1,  find  the  angular  speed  and  acceleration  of  the  radius  • 
drawn  to  the  moving  point. 


IV,  §  53]  RELATED   RATES  89 

12.  A  point  moves  along  the  parabola  y  =i2x-  —  x  in  such  a  manner 
that  the  speed  of  the  abscissa  a;  is  1  ft. /sec.  Find  the  general  expression 
for  the  speed  of  y  ;  and  find  its  value  when  x  =  2  ;  when  x  =  4. 

13.  In  Ex.  12,  find  the  horizontal  and  vertical  accelerations,  the  total 
speed,  the  tangential  acceleration,  and  the  total  acceleration.  [See  Exs. 
10-18,  p.  74.] 

14.  A  point  moves  on  the  cubical  parabola  y  —  t>  in  such  a  way  that 
the  horizontal  speed  is  3  ft./sec.  Express  the  vertical  speed  when  x  =  0. 
Find  its  value. 

15.  Find  the  quantities  mentioned  in  Ex.  13,  for  the  problem  stated 
in  Ex.  14. 

'  16.  If  a  person  walks  along  a  sidewalk  at  the  rate  of  3  mi.  an  hour 
toward  the  gate  of  a  yard,  how  fast  is  he  approaching  a  house  in  the  yard 
which  is  50  ft.  from  the  gate  in  a  line  perpendicular  to  the  walk,  when 
he  is  100  ft.  from  the  gate  ?     When  10  ft.  from  the  gate  ? 

17.  Two  ships  start  from  the  same  point  at  the  same  time,  one  sailing 
due  east  at  10  knots  an  hour,  the  other  due  northwest  at  12  knots  an 
hour.  How  fast  are  they  separating  at  any  time  ?  How  fast,  if  the  first 
ship  starts  an  hour  before  the  other? 

18.  H  a  ladder  8  ft.  long  rests  against  the  side  of  a  room,  and  its  foot 
slips  along  the  floor  at  a  uniform  rate  of  1  ft./sec,  how  fast  is  the  top 

.  descending  when  it  is  6  ft.  above  the  floor? 

'  19.  The  sides  of  a  right  triangle  about  the  right  angle  are  originally 
3  ft.  and  5  ft.  long,  and  grow  at  the  rates  of  3  in.  and  2  in.  a  second,  re- 
spectively. Express  the  lengths  of  these  sides  in  terms  of  the  time  «,  and 
calculate  the  rates  of  change  per  second  of  the  area,  and  of  the  tangents 
of  each  of  the  acute  angles  of  the  triangle.  "Wliat  are  these  rates  when 
t  =  1  sec.  ?  when  t  =  10  sec.  ?     At  what  moment  is  the  triangle  isosceles  ? 

20.  If  the  radius  of  a  sphere  increases  as  the  square  root  of  the  time  ; 
^  determine  the  time-rate  of  change  of  the  surface  and  that  of  the  volume  ; 

the  acceleration  of  the  surface  and  that  of  the  volume. 

21.  Express  the  area  between  the  x-axis  and  the  line  y  =  x  —  1  from 
X  =  1  to  X  =  xo  in  terms  of  xq.  As  Xy  changes  show  that  the  rate  of 
change  of  this  area  is  measured  by  Xf,  —  1  or  y^,. 

22.  If  the  space-time  equation  of  a  motion  is  s=(ffl  +  ^0"'^  show 
that  the  speed  varies  inversely  as  the  tangential  acceleration. 

23.  What  is  the  time-rate  of  change  of  the  force  acting  on  a  body  of 
mass  m  which  moves  on  a  straight  line  with  the  speed  v  =  at-  +  It  -\-  cf 


90  APPLICATIONS   OF   DIFFERENTIATION      [IV,  §  53 

24.  If  a  projectile  is  fired  at  an  angle  of  elevation  a  and  with  muzzle 
velocity  Vq,  its  path  (neglecting  the  resistance  of  the  air)  is  the  parabola 

y  =  X  tan  a ^ , 

2y(,2cos2a' 

X  being  the  horizontal  distance  and  y  the  vertical  distance  from  the  point 
of  discharge.  Draw  the  graph,  taking  g  =  B2,  a  =  20°,  Vq  —  2000  ft. /sec. 
Calculate  dy  in  terms  of  dx.  In  what  direction  is  the  projectile  moving 
when  X  =  5000  ft.,  10,000  ft.,  20,000  ft.  ?     How  high  will  it  rise  ? 

25.  If  in  an  experiment  on  compressing  a  gas  it  is  known  that  pressure 
X  volume  =  constant,  and  the  time-rate  of  change  of  the  pressure  is 
1  +  t^,  calculate  the  time-rate  of  change  of  the  volume ;  compare  the 
acceleration  of  the  pressure  and  that  of  the  volume. 

26.  Ifp-v-  k,  compare  dp/dt  and  dv/dt  in  general ;  compare  d^p/dfi 
and  d-v/dt^. 

27.  If  p  •  -y"  =  ^•,  compare  dp/dt  and  dv/dt.  [For  air,  in  rapid  com- 
pression, n  =  1.41,  nearly.] 

28.  If  q  is  the  quantity  of  one  product  formed  in  a  certain  chemical 
reaction  in  time  «,  it  is  known  that  q  =  ckH/{l  +  ckt).  The  time-rate  of 
change  of  q  is  called  the  speed  v  of  the  reaction.     Show  that 

V  = ^^'       =  cCk  -  qy. 

(1  +  ckty 

Show  also  that  the  acceleration  a  of  the  reaction  is 

2c2F      ^_2cHk-qy. 
{1  +  ckty^ 


CHAPTER  V 

REVERSAL    OF    RATES  —  INTEGRATION  —  SUMMATION 

PART   I.     INTEGRALS   BY   REVERSAL   OF   RATES 

54.  Reversal  of  Eates.  Up  to  this  point,  we  have  been 
engaged  in  finding  rates  of  change  of  given  functions.  Often, 
the  rate  of  change  is  known  and  the  values  of  the  quantity 
which  changes  are  unknown ;  this  leads  to  the  problem  of  this 
chapter:  to  find  the  amount  of  a  quantity  ivhose  rate  of  change  is 
knoivn. 

Simple  instances  of  this  occur  in  every  one's  daily  experience.  Thus, 
if  the  rate  r  (in  cubic  feet  per  second)  at  which  water  is  flowing  into  a 
tank  is  known,  the  total  amount  A  (in  cubic  feet)  of  water  in  the  tank 
at  any  time  can  be  computed  readily,  —  at  least  if  the  amount  originally 
in  the  tank  is  known : 

A^r-t  +  C, 

where  t  is  the  time  (in  seconds)  the  water  has  run,  and  C  is  the  amount 
originally  in  the  tank,  i.e.  C  is  the  value  of  A  at  the  time  when  t  =  0. 

If  a  train  runs  at  30  miles  per  hour,  its  total  distance  d,  from  a  given 
point  on  the  track,  is 

d  =  30.t+  C, 

where  t  is  the  time  (in  hours)  the  train  has  run,  and  C  is  the  original 
distance  of  the  train  from  that  point,  i.e.  C  is  the  value  of  d  when  t  =  0. 
(Notice  that  by  regarding  d  as  negative  in  one  direction,  this  result  is 
perfectly  general ;   C  may  also  be  negative.) 

If  a  man  is  saving  $  100  a  month,  his  total  means  is  100  •  ;i  +  C,  where 
n  is  the  number  of  months  counted,  and  C  is  his  means  at  the  beginning  ; 
i.e.  C  is  his  means  when  n  —0. 

If  the  cost  for  operating  a  printing  press  is  0.01  ct.  per  sheet,  tlie  total 
expense  of  printing  is 

T  =  0.01  ■  n  +  O 
91 


92  INTEGRATION  [V,  §  54 

where  n  is  the  number  of  copies  printed,  and  where  C  is  the  first  cost  of 
the  machine  ;  i.e.  C  is  the  value  of  T  when  n  =  0. 

55.  Principle  Involved.  Such  simple  examples  require  no 
new  methods ;  they  illustrate  excellently  the  following  fact : 

Tlie  total  amount  *  of  a  variable  quantity  y  at  any  stage  is  deter- 
mined ichen  its  rate  of  iyicrease  and  its  original  value  C  are  known. 

We  shall  see  that  this  remains  true  even  when  the  rate 
itself  is  variable. 

56.  Illustrative  Examples.  The  rate  R{x)  at  which  any 
variable  y  increases  with  respect  to  an  independent  variable 
X  is  the  derivative  dy/dx ;  hence  the  general  problem  of  §  54-55 
may  be  stated  as  follows  :  given  the  derivative  dy/dx,  to  find  y 
in  terms  of  x. 

In  many  instances  our  familiarity  with  the  rules  for  obtain- 
ing rates  of  increase  (differentiation)  enables  us  to  set  down  at 
once  a  function  which  has  a  given  rate  of  increase. 

Example  1.  Thus,  in  each  of  the  examples  given  in  §  54,  the  rate  is 
constant ;  using  the  letters  of  this  article  : 

^^  =  Bix)  =  lc, 
dx 

where  A;  is  a  known  fixed  number ;  it  is  obvious  that  a  function  which 

has  this  derivative  is 

{A)  y  =  kx+  C, 

where  C  is  any  constant  chosen  at  pleasure. 

While  the  examples  of  §  54  can  all  be  solved  very  easily  without  this 
new  method,  for  those  which  follow  it  is  at  least  very  convenient.  The 
value  of  C  in  any  given  example  is  found  as  in  §  64 ;  it  represents  the 
value  of  y  when  x  =  0. 

Example  2.     Given  dy/dx  —  .r^,  to  find  y  in  terms  of  x. 

Since  we  know  that  d(x^)/dx  =  3  x^,  and  since  multiplying  a  function 
by  a  number  multiplies  its  derivative  by  the  same  number,  we  should 
evidently  take  : 

*  This  total  amount  is  what  is  called  in  §  57  the  integral  of  the  rate  ;  the 
word  integral  means  precisely  the  "  total "  made  from  the  rate,  by  its  English 
derivation  ;  compare  the  English  words  entire,  entirety,  integrity,  integer,  etc. 


V,  §  56]  REVERSAL  OF  RATES  93 

i/=:-,    or  else   y^^  +  C;    Tcheck  :   d  (^  +  c\  =  x'^dx]  , 

where  C  is  some  constant.  As  in  §  54,  some  additional  information  must 
be  given  to  determine  C.  In  a  practical  problem,  such  as  Ex.  3,  below 
information  of  this  kind  is  usually  known. 

Example  3.  A  body  falls  from  a  height  100  ft.  above  the  earth's  sur- 
face ;  given  that  the  speed  is  u  =  —  gt,  find  its  distance  from  the  earth 
in  terms  of  the  time  t. 

Let  s  denote  the  distance  (in  feet)  of  the  body  from  the  earth  ;  we  are 
given  that 

(1)  V  =  —  =  —  gt,    or  ds  -  vdt  =  —  gt  dt, 

which  is  negative  since  s  is  decreasing.  We  know  that  d(t^)  =  2tdt; 
hence  it  is  evident  that  we  should  take  : 

(2)  s  =  ~^f^+  C;    [check  :  ds  =  -  gt  dt]. 

As  the  body  starts  to  fall,  t  =  0  and  s  =  100  ;  substituting  these  values 
in  (2)  we  find  100  =  0  +  C,    or    C  =  100. 

In  this  problem,  therefore,  we  have 

s  =  _  2^2  +  100. 

Example  4.     Given  dy/dx  =  x",  to  find  y  in  terms  of  x. 
Since  we  know  that  d(a;"+i)  =  {n  +  \)x"dx.,  we  should  take 

(£)  y  =  —i— x"+i  +  C ;    [check  :  dy  =  x" dx]. 

71  +  1 

Since  the  rule  for  differentiation  of  a  power  was  proved  (§  23,  p.  38) 
for  all  positive  and  negative  values  of  n,  the  formula  (i?)  holds  for  all 
these  values  of  n  except  n  =  —  \;  when  n  =  —  1  the  formula  {B)  cannot 
be  used  because  the  denominator  ji  +  1  becomes  zero.     (See  §  78,  p.  136.) 

Special  cases : 
i'  „_i    #_„    "^la;2+ C;  check:  (Zf-5c2'\=xcZx. 

X  -{■  C ;  check  :  d{x)  =  1  •  dx. 

di^xy2]  =  xyHx. 


=  1 

dx 

=  0 

dx 

_1 

2 

dx 

=  - 

dx 

=  - 

1     drj 
3'  dx 

'M 


x~^,   y  =  —  x~'  4-  C ;  check  :  d{—  l  a—')  =  x--dx. 

a;  1/3    y  z=  ?a;2/3  ^  c ;  check  :  di~x^'A  =  x-^^dx. 
'   '      2  '  \2       I 


94  INTEGRATION  [V,  §  56 

Notice  that  these  include  Vx(=  a;'/^),  \/x^(=  x~^),  etc.  ;  other  special 
cases  are  left  to  the  student. 

Example  5.     Given  dy/dx  =x^  +  2  x^,  to  find  y  in  terms  of  x. 

Since  d(x'^)/dx  =  4  x^  and  d{x^)/dx  —  3  x^,  and  since  the  derivative  of 
a  sum  of  two  functions  is  equal  to  the  sum  of  their  derivatives,  it  is 
evident  that  we  should  write  * 

The  check  is 

#  =  Af^+2^+cUx3  +  2x2; 
dx      dx\4:         3  / 

such  a  check  on  the  answer  should  be  made  in  every  exercise. 

In  general,  as  in  this  example,  if  the  given  rate  of  increase  (derivative) 
is  the  sum  of  two  parts,  the  answer  is  found  by  adding  the  answers 
which  would  arise  from  the  parts  taken  separately,  since  the  sum  of  the 
derivatives  of  two  variables  is  always  the  derivative  of  their  sum. 

EXERCISES  XIX.— REVERSAL  OF  RATES 

1.  Determine  functions  whose  derivatives  are  given  below  ;  do  not 
forget  the  additive  constant ;  check  each  answer. 

Ca)^  =  ix.         (6)^  =  -5x.        (c)^^  =  3x2.        (d)^^  =  2. 
^  ■*  dx  ^      ilx  ^  ^  dx  ^  '  dx 

^  ^  dx  ^^  dx  ^^^  dx  ^      dx 

2.  In  the  following  exercises,  remember  that  the  derivative  of  a  sum 
is  the  sum  of  the  derivatives  of  the  several  terms  ;  proceed  as  in  Ex.  1. 

(a)  ^  =  4  +  5x2.         (6)^=4x2-2x4-3.     (c)  ^  =  «3  -  4«  +  7. 
dx  dx  dt 

(d)^  =  Sx5-8x\       (e)^=ax  +  b.  (f)  -  =  at'' +  bt  +  c. 

dx  dx  dt 

I 

(^)  ^  =  .006 x2  -  .004  x3  +.015 X*.       (h)  ~  = -t^  +  bt^  -  Gt'^  +  2.    ^ 
dx  dt 

(i)  ^  =  x-3.  (j)   -  =  t^  + 1/«2.  (^•)  ^  =  3  r2  +  4  r8. 

dx  dt  dt 

(l)   ^  =  x2/3.  (m)    '^y-  =  2  xi/2  _  3  x-i  /2.     („)  ^  ^  kv-2-*\ 

dx  dx  dv 

*  In  all  the  Examples  of  this  paragraph,  we  have  had  an  equation  which 
Involves  dy/dx ;  such  an  equation  is  often  called  a  differential  equation,  be« 
cause  it  contains  differentials.    See  also  Chapter  X. 


V,  §  56]  REVERSAL   OF  RATES  95 

3.  As  a  train  leaves  a  station,  its  speed  v  is  proportional  to  the  time  ; 
tind  the  relation  between  the  distance  s  passed  over  and  the  time. 

[Hint,     v  =  ds/dt  =  kt.     Here  and  below,  the  unit  of  time  is  1  sec.] 

4.  If  in  Ex.  3,  A =1/4,  find  s  when  t  =  10.  What  is  the  average  speed 
during  this  time  ?  Is  the  actual  speed  ever  equal  to  this  average  speed  ? 
When  ?     Try  to  make  a  rough  estimate  in  advance. 

5.  Compare  the  speeds  and  the  distances  passed  over  by  an  express 
train  which  leaves  a  station  with  an  increasing  speed  v=t/2,  with  that 
of  a  freight  train  which  stai'ts  from  a  point  100  yd.  ahead  at  the  same 
in.stant  with  a  speed  v  =  t/lO. 

6.  Determine  v  and  s  in  terms  of  t  for  a  bullet  shot  vertically  upward 
with  a  speed  2000  ft. /sec. 

[Hint.  Acceleration  =dv/d<  =  —  ^=—32.2  ft. /sec. /sec. ;  v  =  2000  when 
^  =  0  ;   s  =  0  when  t  =  0.     Neglect  the  air  friction.] 

7.  How  high  will  the  bullet  in  Ex.  6  rise  ?  How  long  will  it  remain 
in  the  air  ?    Make  a  rough  estimate  in  advance. 

8.  A  car  starts  with  a  speed  v  =  t^/l2  ;  find  s  ;  how  far  will  it  go  in 
3  seconds  ? 

9.  A  flywheel  starts  with  an  angular  speed  w  =  .01  t-  in  radians  per 
second.  How  long  does  it  take  to  make  the  first  revolution  ?  How  long 
for  the  next  ? 

10.  If  a  flywheel  starts  with  a  speed  w  =  .001 1^,  what  is  the  time  of 
the  first  revolution  ?  of  the  second  ?  of  the  tenth  ? 

11.  If  the  angular  speed  in  radians  per  second  of  a  wheel  while  stop- 
ping is  w  =  100  —  10 1,  how  many  revolutions  will  it  make  before  it  stops  ? 

12.  Determine  the  form  of  the  surface  of  water  in  a  rapidly  rotated 
bucket  from  the  fact  that  any  vertical  section  through  its  lowest  point 
has  a  slope  dy/dx  =  (o}'^/g)x,  where  x  is  measured  horizontally  and  y 
vertically  from  the  lowest  point,  w  is  the  angular  speed  in  radians  per 
second,  and  g=^2.2.     Plot  the  section  when  w  =  8. 

13.  Determine  a  curve  through  (0,  0)  whose  slope  is  proportional  to 
z;  to  x2  ;  to  1  —  x^. 

14.  Determine  a  curve  through  (0,  0)  and  (1,2)  whose  flexion  is  pro- 
portional to  X  ;  to  1  -1-  x2  ;  one  whose  flexion  is  constant. 

15.  Determine  the  form  of  the  upper  surface  of  a  beam  if  its  flexion  is 
constant,  and  if  tlie  beam  rests  on  two  fixed  supports  at  a  distance  I 
from  each  other.     See  Ex.  20,  p.  80. 


96  INTEGRATION  [V,  §  57 

57.  Integral  Notation.  If  the  rate  of  increase  dy/dx  =  R{x) 
of  one  variable  y  with  respect  to  another  variable  x  is  given,  a 
function  y  =.  I (x)  tvhich  has  i^recisely  this  given  rate  of  increase 
is  called  an  indefinite  integral  *  of  the  rate  li  (x),  and  is  repre- 
sented by  the  symbol  t 

(1)  !(•«)=  ('jR{:Jc)dx; 

that  is, 

<^2)  *^      ^l^-^(-*')J=^(^)'         then  I(x)=j'll(x)dac, 

or,  vyhat  amounts  to  the  same  thing, 

(3)  if  d\_Iix)^  =  B(x)dx,   then    I(x)=    CB(x)dx. 

The  results  of  Examples  1,  2,  3,  §  56,  written  with  the  new 
symbol,  are,  respectively, 

[^]  Ck  dx  =  kx  +  C. 

jx'dx  =  x^S  +  a 
s  =  Jy  dt+C  =  j~ gt  dt  +  C=  -  gt-/2  +  C. 
The  first  equation  of  Example  3  holds  in  general : 


[I]  s  =/^ 


dt  +  C,    since    —  =  v. 
dt 


The  result  obtained  in  (B),  Example  4,  §  56,  gives 
jc^dx  =  '^ +C,    n^-1, 

71  +  1 

*  The  common  Eno;lish  meaning  of  the  word  integrate  is  "to  make  whole 
again,"  "  to  restore  to  its  entirety,"  "to  ^ive  the  sum  or  total."  See  any 
dictionary,  and  compare  §§  54-55. 

To  integrate  a  rate  R{z)  is  to  find  its  integral;  the  process  is  called  integra- 
tion. Often  the  rate  function  Ii{x)  which  is  integrated  is  called  the  integrand ; 
thus  the  first  part  of  equation  (2)  may  be  read  :  "  the  derivative  of  the  integral 
is  the  integrand."     This  is  the  property  used  in  checking  answers. 

The  first  equation  in  (2)  and  the  first  in  (3)  are  differential  equations.  See 
footnote,  p.  94. 

t  Note  that  dx  is  part  of  the  symbol.  As  a  blank  symbol,  it  is  /  (blank)  dx ; 
the  function  R(x)  to  be  integrated  {i.e.  the  integrand)  is  inserted  in  place  of 
the  blank.    The  origin  of  this  symbol  is  explained  in  §  67,  p.  117. 


V,  §  57]  NOTATION  97 

for  all  positive  and  negative  integral  and  fractional  values  of  n 
except  11  =  — 1,  for  which  see  §  78,  p.  13G. 

As  examples  of  the  many  special  cases,  we  write : 

«  =  1,         (xdx  =^  +  C. 

n  =  0,         (xMx,  =(iax.      =  ffZx  =x+C. 

n  =  \,  ^x^'-dx  =  r  Vx  dx  =  f  x^'-  +  C=  f  \/^+  C. 

M  =  -  2,     (xr-dx  =  (\  dx  =  -  x"^  +  C  =  --+  O. 

n  =  -  1      r.v-i/''rfx  =  ^~dx  =  f  x2/3  +  C  =  f  Vx2  +  C. 
'^      ^  '^  Vx 

From  Example  5 : 

r  (..-3  +  2  x2)dx  =  (xMx  +  (-2  x^dx        ==  ^  +  ?-£!+  C. 

The  general  principle  used  in  this  example  is  that  the  inte' 
gral  of  a  sum  of  two  functions  is  the  sum  of  their  integrals  : 

[C]  fciJCx)  4-  -SCx)]  (Ijc  =  i  B,{x)  dx  +  f-SCir)  dx, 

which  is  true  because  the  derivative  of  the  sura  R(x)  +  S{x)  is 
the  sum  of  the  derivatives  : 

dlR  +  S^/dx  =  dli/dx  +  dS/dx. 

The  rules  {A),  (B),  (C)  are  sufficient  to  integrate  a  large 
number  of  functions,  including  certainly  dM  x>objnomials  in  x. 

EXERCISES    XX.  — NOTATION  — INDEFINITE   INTEGRALS 

1.  Express  the  value  of  y  if  dy/dz  =  4x''^  +  3x  by  means  of  the  new 
sign  /  ( )dx.     Also  find  y.     Check. 

2.  If  dy/dx  has  any  one  of  the  following  values,  express  y,  first  by  use 

of  the  new  sign  J  ( )dx  and  then  directly  in  terms  of  x.     Check  the 

final  answers.     Do  not  omit  the  arbitrary  constant. 

(a)  x2.     (c)  X*.  (e)   x2  -  2.         (g)  xP-  -  2  x  +  3.      (0   ax  4-  h. 

(6)  x8.     (d)2x  +  3.     (/)  7.  (K)x?-x\  (j)Vi. 

H 


98  INTEGRATION  [V,  §  57 

3.  In  many  examples  it  is  profitable  first  to  expand  the  given  expres- 
sion in  a  sum  of  powers ;  proceed  as  in  Ex.  2,  and  find  y  if  dy/dx  has 
any  of  the  following  values  : 

(a)x(l+x).  (e)    (l  +  a;2)(l-x2).  (i)    (\  +  x)  (\  -  x-^) . 

[b)  (x3  +  4  a;2) -4- X.     (/)   (2  -  3x)(4  +  x).  (j)    x'/s(x  +  x^). 

[c)  4(x  +  2)2.  (fj)  xi/-'(l  +  aO-  W   (x3  +  2  x2  +  x)V2. 

[d)  2x2(3-4x2).       (h)   (l-ix)Vx.  (0   (a;2  +  2)2(2xV2  +  3). 

4.  Integrate  the  following  expressions  : 
rt)    ijx^dx.  (d)    ij(l  +  v)dv.  (g)     ( xr^dx. 

b)  (sthlt.  (e)    r(3x2-4x3)dx.  (h)     (sr^dt. 

c)  (is-^ls.         (/)(*(  10-9  x-3)dx.  (0      Cfl/--dt. 

j)  r  (3  M  +  5  ?«2  +  7  u^)au.  (k)     (  (7/x^2  +  8  x-io  -  10/x'')(Zx. 

Z)  r2^x2dx.       C"0    r(5s*-3s2  +  2)ds.      (?i)     fCits/s  +  10<5/2)  (^^. 

.    Integrate  the  following  expressions,  making  use  of  the  principle  of 
3: 

a)  ((1-tydt.  (g)   ^Vx(a  +  bx)dx. 

b)  (x{l  +  Vx)dx.  (/i)    (x''(a  +  bx)dx. 

c)  rs(l-v^)2ds.  (0    j'(a  +  6x)2dx. 

d)  (t%l-t'-)dt.  U)      j'(<2'5-2)«-5(?«. 

e)  rx-i(l  +  X  +  x2)  (te.  (k)    (x^*0  -{-x'^ydx. 
y)    rx(ff  +  6x)fZx.                                (?;>     rV«(l  +  2t-y-^dt. 

6.  Powers  of  linear  expressions  may  be  treated  without  expanding. 
Find  a  function  whose  derivative  is  (x  +  1)2  by  analogy  with  the  function 
whose  derivative  is  x2. 

7.  Can  y  be  found  when  dy/dx  =  (x  +  3)2  by  the  same  analogy  ?  Can 
y  be  found  when  dy/dx  =  (2  x  +  3)2  ?  Be  sure  to  check  your  answers. 
If  they  are  wrong,  put  in  the  proper  factor  to  correct  the  error. 

8.  Find  y  when  dy/dx  has  one  of  the  following  values : 

(a)  (3x-2)3.  (6)   (2x  + 1)1/2.  (c)   (5x-4)-3. 


V,  §  58]  FUNDAMENTAL  THEOREM  99 

58.  Fundamental  Theorem.  We  have  seen  that  such  func- 
tions as  x^,  OCT  +  o,  X-  4-  C,  where  C  is  any  constant,  have  the 
same  derivative  2  x.  If  the  rate  of  increase  (derivative)  of  y 
with  respect  to  x  is  given,  there  may  be  several  answers  for 
y  in  terms  of  x ;  thus  if  dy/dx  =  2x,  the  answers  y  =  x^, 
y  =  3r-\-o,  y  =  X-  +  C  are  all  correct  solutions;  to  decide  whicli 
one  is  wanted,  additional  information  is  needed,  as  in  §  54 
and§  56  (Exs.  2,3,  etc.). 

However,  except  for  the  additive  constant  C,  all  answers 
coincide;  for  practical  purposes,  there  is  but  one  answer. 
Stated  precisely,  this  is  the  Fundamental  Theorem  of  Integral 
Calculus : 

If  the  rate  of  increase 

(1)  |  =  7e(.) 

of  a  variable  quantity  y  ivhich  depends  on  x  is  given,  then  y  is 
determined  as  a  function  of  x,  I(x),  except  for  a  constant  term: 

(2)  y  =^R{x)dx  +C=I(x)+  a 

Stated  in  different  words  this  theorem  is : 
T7ie  difference  betiveen  any  two  functions  I{x)  and  J{x)  ivhose 
derivatives  are  equal,  is  a  constant. 

Let  this  difference  be 

D(x)  =  I(x)-J(x); 
then  dD{x)^dI(x)      dj(x)  ^^^ 

dx  dx  dx 

Since  the  rate  of  increase  (derivative)  of  D(x)  is  zero,  D(x) 
neither  increases  nor  decreases  for  any  value  of  x;  hence  D(x) 
is  a  constant,  as  was  to  be  proved.* 

The  constant  C  which  occurs  in  the  answers  is  always  to  be 
determined  by  additional  information,  as  in  §  54  and  §  56. 

♦Graphically,  the  "curve"  which  represents  I>{x)  has  its  tangent  hori- 
zontal at  every  point,  —  such  a  "curve"  is  necessarily  a  horizontal  straight 
line  :  D(x)=  constant.     (See  also  §  131.) 


100  INTEGRATION  [V,  §  59 

59.  Definite  Integrals.  In  applications,  we  often  care  little 
about  the  actual  total ;  it  is  rather  the  difference  between  two 
values  which  is  important. 

Thus,  in  a  motion,  we  care  little  about  the  real  total  distance 
a  body  has  traveled  since  the  creation  of  the  universe ;  it  is 
rather  the  distance  it  has  traveled  between  two  given  instants. 

If  a  body  falls  from  any  height,  the  distance  it  falls  is  (Ex.  3,  p.  93) 

s^(vdt+  C  =  ^gtdt+C  =  ^+  C, 

■where  s  is  counted  downwards. 

The  value  of  s  when  f  =  0  is  s]«=o  =  C ;  the  value  of  s  when  f  =  1  is 
s']i=i  =  g/2  4-  C.  The  distance  traversed  in  the  first  second  is  found  by 
subtracting  these  values  : 


=0         J<=1         J«=o       \^ 


C'j-C  =  ^  =  16.1ft., 


where  s~\  'l!  means  the  space  passed  over  between  the  times  t  =  0  and  t  =  l. 

In  this  calculation,  we  care  little  about  where  .s  is  counted  from ;  or 
its  total  value.  The  result  is  the  same  for  all  bodies  dropped  from  any 
height. 

Likewise,  the  space  passed  over  between  the  times  t  =  '2  and  f  =  5  is 

^g^_25_g^  =^.21  ^338  (ft.). 
In  general  the  distance  traversed  between  the  times  t  =  a  and  t  =  b  is 


i;::-L-a=.=("f 


The  advantage  realized  in  this  example  in  eliminating  C 
can  be  gained  in  all  problems : 

TJie  numerical  value  of  the  total  change  in  a  qtiantity  between 
two  values  of  x,  x  =  a  and  x  =  b,  can  be  found  if  the  rate  of 
change  dy/dx  =  B{x)  is  given.     For,  if 


y  =  I(x)=  (R(x)  dx  +  C, 


V,  §  59]  DEFINITE   INTEGRALS  101 

the  value  of  y  for  x  =  a  is 

and  the  value  of  y  for  x  =  h  is, 

The  total  change  in  y  between  the  values  x  =  a  and  x  =  6  is 

This  difference,  found  by  subtracting  the  values  of  the  indefinite 
integral  at  x  =  a  from  its  value  at  x  =  b,  is  called  the  definite 
integral  of  R{x)  between  x=a  and  x  =  b  ;  and  is  denoted  by  the 
symbol  : 

J        It(x)dx  =  \j  Kijc)djc\    -\jlt(x)ax\ 

=  1(b) -1(a). 

It  should  be  noticed  that,  in  subtracting,  the  unknown  con- 
stant C  has  disappeared  completely ;  this  is  the  reason  for 
calling  this  form  definite. 

Example  1.     Given  dy/dx  =  x^,  fiud  the  total  change  in  y  from  x  =  1  to 
x  =  3. 
Since 

y  =  (x^  dx  =  x*/4:  +  C, 
it  follows  that 

,]•-=,]    -,]       =fl    -fl       =20. 

Ji=l  Jx=3      Jx=l         4Jx=3    4Jx=l 

Interpreted  as  a  problem  in  motion,  where  x  means  time  and  y  means 
distance,  this  would  mean  :  the  total  distance  traveled  by  a  body  be- 
tween the  end  of  the  first  second  and  the  end  of  the  third  second,  if  its  speed 
is  the  cube  of  tbe  time,  is  twenty  units. 

Interpreted  graphically,  a  curve  whose  slope  m  is  given  by  the  equation 
TO  =  x^  rises  20  units  between  x  =  l  and  x  =  3.  The  equation  of  the 
curve  is  ?/  =  x^/i  +  C. 


102  INTEGRATION  [V,  §  59 


EXERCISES   XXI— DEFINITE  INTEGRALS 

1.  If  water  pours  into  a  tank  at  the  rate  of  200  gal.  per  minute,  how 
much  enters  in  the  first  ten  minutes  ?  how  much  from  the  beginning  of 
the  fifth  minute  to  the  beginning  of  the  tenth  minute  ? 

2.  If  a  train  is  moving  at  a  speed  of  20  mi.  per  hour,  how  far  does  it 
go  in  two  hours  ?  Does  this  necessarily  mean  the  distance  from  its 
last  stop  ? 

3.  If  a  train  leaves  a  station  with  a  variable  speed  v  =  1/4  (ft./sec), 
find  s  in  terms  of  t.  How  far  does  the  train  go  in  the  first  ten  seconds  ? 
How  far  from  the  beginning  of  the  fifth  to  the  beginning  of  the  tenth 
second  ? 

4.  A  falling  body  has  a  speed  v  =  gt^  where  t  is  measured  from  the 
instant  it  falls.  How  far  does  it  go  in  the  first  five  seconds  ?  How  far 
between  the  times  t  =  S  and  t  =  7  ? 

5.  A  wheel  rotates  with  a  variable  speed  (radians/sec.)  w  =  t-/lOO. 
How  many  revolutions  does  it  make  in  the  first  fifteen  seconds  ?  How 
many  between  the  times  i  =:  5  and  t  =  20? 

6.  From  the  following  rates  of  change  determine  the  total  change  in 
the  functions  between  the  limits  indicated  for  the  independent  variable. 
Interpret  each  result  geometrically  and  as  a  problem  in  motion,  and  write 
your  work  in  the  notation  used  in  the  text : 

(a)  ^  =  x,  x  =  1  to  X  =  2. 
dx 

(ft)  ^  =  i  a:2,  a;  =  -  2  to  ic 

(0 

(^d)  !i^  =  1  -  x2,  X  =  0  to  X  =  10. 


dx 

Ax 
4 

dy_ 

X3 

dx 

12 

dl__ 
dx 

=  1  - 

(/) 

ds  _ 
dt  ' 

_«i-2 
t^    ' 

t  = 

1  to  e  =  3. 

(9) 

dv  ^ 
dt  ' 

_(l  +  0- 

(3/2 

,  i 

=  16tot  =  25. 

(h) 

dv_ 
dJt 

_  2  ri  - 
t- 

Sr 

-,<  =  0.1to0.01 

(0 

de_ 

dt 

-Wtvi 

t 

=  0  to  «  =  .01. 

U) 

dd  _ 
dt  ~ 

_  a^  _  a^ 

t  - 

=  a  to  «  =:  2  rt. 

-,t  =  lOtot  =  100. 
dt      f^ 

7.  Determine  the  values  of  the  following  definite  integrals.  [In  cases 
where  no  misunderstanding  could  possibly  arise,  only  the  numerical  values 
of  the  limits  are  given.  In  every  such  case,  the  numbers  stated  as  limits 
are  values  of  the  variable  whose  differential  appears  in  the  integral.] 


V,  §  60] 


AREA  UNDER  A  CURVE 


103 


(a) 

i:> 

dx. 

(6) 

s:> 

xdx. 

(0 

s:> 

C2  dx. 

(d) 

o 

x^dx. 

(»>  X* 


Vxdx. 


(/)    j*^V  3  dx.  (k)      (    V/3(s2-2  S)  dS. 
/•r=(1  /•  0=100 

(/i)    i_2((-^ -!)(/(  (m)  j     ^^   (.01  +  .02^)d<?. 

(0  J_'2  ( 1  +  s + s'^)  ds.  ( « )  1^'' {^e  +  -h-)  ^^■ 

0-)  f  ^*^-  («)  ^y-'"'^^- 


8.  A  stone  falls  with  a  speed  v  =  gt  +  10.  Find  s  in  terms  of  t  and 
find  the  distance  passed  over  between  the  times  t  =  2  and  t  —  7. 

9.  A  bullet  is  fired  vertically  with  a  speed  v  =  —  gt  +  1500.  How  far 
does  it  go  in  ten  seconds  ?  How  high  does  it  rise  ?  How  long  is  it  in  the 
air  ?     Make  rough  estimates  of  the  answers  in  advance. 

10.  For  any  falling  body,  j  =  acceleration  =  g  =  const.  Find  the 
increase  in  speed  in  ten  seconds.  Does  it  matter  what  particular  ten 
seconds  are  chosen  ? 

11.  If,  in  Ex.  10,  the  speed  is  100  ft. /sec.  when  t  =  5,  what  is  the  speed 
when  f  =  15  ?  When  will  the  speed  be  250  ft./sec.  ?  Express  v  in  terms 
of  t. 

60.  Area  under  a  Curve.  Consider  the  area  A  under  any 
curve  y  =/(.r),  between  the  .r-axis  and  the  curve,  and  between 
a  fixed  vertical  line  through  a 
fixed  point  F,  (x  =  k),  and  a  mov- 
able vertical  line  through  a  mov- 
able point  M,  (x  =  x). 

As  3f  moves  to  N,  x  increases 
by  an  amount  Ax  =  MN,  and  A  in- 
creases by  an  amount  A^  =  area 
of  MNRQ.     The  average  rate  of  ^'°-  -• 

increase  of  A  is  A  A  ~  Ax,  which  is  equal  to  some  height  be- 
tween the  extremes  of  the  values  of  y  along  MN.  As  Ax  ap- 
proaches zero,  this  intermediate  height  approaches  the  height 
at  M;  and  the  instantaneous  rate  of  increase  in  A  is 


104 


INTEGRATION 


[V,  §  60 


dx       Ax-o  Ao;       M.y^    MN  ^ 

the  rate  of  increase  of  A  ivith  respect  to  x  is  the  height  of  the  curve. 

It  follows  that 
(2)  A=j>jdx+C  =  jf{x)dx+C; 

and  the  area  A  between  any  two  fixed  values  of  a:,  a;  =  a  and 
x  =  b  is  the  definite  integral : 


(3) 


^i:>^.j^LrX:' 


f(x)  dx. 


Example  1.  To  find  the  area  under  the 
curve  *  y  —  x^  between  the  points  where 
x  =  0  and  x  =  2. 

We  have,  by  (2) 

A^(ijdx  +  C=  (x^  dx  +  C  =  -  +  C, 

where  A  is  counted  from  any  fixed  back 
boundary  x  =  A;  we  please  to  assume,  up  to 
a  movable  boundary  x  =  x. 

The  area  between  x  =  0  and  x  =  2  is  given 
by  subtracting  the  value  of  A  for  x  =  0  from 
the  value  of  A  for  x  =  2  : 

aT"  =  A-]     -a-]       =j-'.=.x  =  f]     -f]       =|. 

Likevdse  the  area  under  the  curve  between  x  =  1  and  a;  =  3  is 

-j:;:=r:'^'-=f].7'i'i=.-- 

and  the  area  under  the  curve  between  any  two  vertical  lines  x  =  a  and 
X  =  6  is 

b^  —  a^ 


L_ 

1                 2/.                   '- 

i         ^ 

t        y^l 

A              t 

^          2        -4 

I         7t 

^  1  -.tr 

K      7 

K^"^ 

if  0        1      ^ 

it      It: 

Fig.  23. 


♦The  phrase  "the  area  under  the  curve"  is  understood  in  the  sense  used 
in  the  first  sentence  of  §60.  When  the  curve  is  below  the  x-axis,  this  area  is 
counted  as  negative. 


V,  §  60J  AREA  UNDER  A  CURVE  105 

EXERCISES  XXn.  — AREAS 
[Draw  a  figure  and  estimate  the  answer  in  advance,  whenever  possible.] 

1.  Find  the  area  under  each  of  the  following  curves  between   the 
ordinates  x  =  0  and  x  =  1  ;  between  x  =  2  and  x  =  5  : 

(a)y^3x\         (f0    2/=V7._  (r/)  x-'^/ =  1.  (See  §  111.) 

(6)  2/  =  x3.  (e)    2/  =  l/Vx.  (See  §111.)    (h)  y  =  x^  +  -Sx-i. 

(c)    2/  =  xV10.     (/)y=(l-x2).  (0    t/  =  x(l-x)2. 

2.  Find  the  area  between  the  line  y  =  2x  and  the  parabola  y  =  x^. 

3.  Find  the  area  between  y  =  x  and  y  =  s/x. 

4.  Show  that  y  =  x"^  and  j/^  =  x  trisect  the  unit  square  whose  diago- 
nal joins  the  points  (0,  0)  and  (1,  1). 

5.  Find  the  area  between  y  =  x-  and  y  =  x^ ;  and  show  that  it  is  the 
same  as  that  under  the  curve  y  =  x-  —  x^. 

6.  Find  the  areas  under  each  of  the  following  curves  : 

(a)  2/  =  x3  +  6  x2  +  15x,  (x  =  0  to  2  ;  x  =  -  2  to  +  2  ;  x  =  -  a  to  +  a). 
(6)   y  =  x'^'^,  (x  =  0  to  8  ;  X  =  -  1  to  +  1  ;  X  =  —  a  to  +  a), 
(c)   y  =  X-  +  1/-k'-^,  (x  =  1  to  3 ;  x  =  2  to  5 ;  x  =  a  to  6). 

7.  Find  the  area  A  under  the  line  y  =  2x  +  3  between  x  =  0  and 
X  =  X  (any  value)  by  geometry  ;  show  directly  that  dA/dx  =  y. 

8.  Find   geometrically  the   area   A   under   the    line   x  +  y  +  2  =  0 
between  x  =  0  and  x  =  x  ;  show  directly  that  dA/dx  =  y. 

9.  Show  that  the  area  A*  bounded  by  a  curve  x  =  ^(y)  the  j/-axis, 
and  the  two  lines  y  —  a  and  y  =  bi& 

Xy=h 
^Hy)dy. 

10.  Calculate  the  area  between  the  »/-axis,  the  curve  x  =  y"^,  and  the 
lines  2/  =  0  and  y  =  I.     Compare  this  answer  with  that  of  Ex.  3. 

11.  Find  the  area  between  the  curve  y  —  x^  and  each  of  the  axes  sepa- 
rately, from  the  origin  to  a  point  (jfc,  k^).    Show  that  their  sura  is  k*. 

12.  Find  the  area  in  the  first  quadrant  between  y  =  x^  +  3x,  the  j/-axis, 
and  the  lines  ?/  =  0,  ?/  =  4,  by  subtracting  from  a  certain  rectangle  the  area 
between  y  =x^  +  3  x,  the  x-axis,  and  the  lines  x  =  0,  x  =  1. 

13.  Find  the  area  in  the  first  quadrant  between  the  curve  y=x^+2  x— 7, 
the  2/-axis,  and  the  lines  ?/  =  2,  ?/  =  9,  by  the  method  of  Ex,  12. 


M 

Ax 

t" 

/ 

//                    ^ 

K 

>     0 

x=k             x-x     x=x+Ax 

Fig.  24. 

106  INTEGRATION  [V,  §  61 

61.    Lengths  of  Curves.     Let  s  represent  the  length  of  the 
arc  FM  of  a  curve  C  whose  equation  is  y=f(x),  between  a 
Q        fixed    point   F    and   a   moving 
J^y  point  M,  on  the  curve. 

As  the  point  ikf  moves  on  to  N', 
the  value  of  x  increases  by  an 
amount  Ao;  =  JK  =  ML,  y  by 
an  amount  Ay  =  LN,  and  s  by 
an  amount  As  =  arc  MN. 
The  chord  MN  is  given  by  the  Pythagorean  theorem  : 

(1)  [chord  MNy  =  ML^  +  LN"  =  Ax  +  Ky'. 

The  instantaneous  rate  of  increase  of  the  arc  s  with  respect  to 
X  is 

/ON  ds      T      As      V     arc  MN 

(2)  —  =  lim  —  =  lim •  ; 

dx      Ax=^  Ax      Ai=y)      Ao; 

this  limit  can  be  found  by  the  fundamental  fact  (§  12)  that  the 
limit  of  the  ratio  of  an  arc  to  its  subtended  chord  is  unity  * 

,os  ds     T     arc  MN      ■,■     chord  MN 

(3)  — =lim =  lim , 

dx     Ax=y)      Ax  Ax^         Ax 

,.         arc  MN       ^ 
MN=o  chord  MN 
hence 

(4)  W  =  [lim  ("I'o-l^yn  =  ,i,„  g  +  ^ 


Ax^|_        \Ax)  ]i' 


Ax 


{^■^) 


or 


It  follows  that  the  total  are  is 

(6)  s=f^jl  +  (^ydcc  +  C=J^^+7It'dx+C, 

*  This  fact  is  the  pith  of  the  argument ;  it  contains  the  essence  of  the  defi- 
nition of  what  we  mean  by  the  length  of  an  arc  of  a  curve.    See  §  12. 


V,  §  62] 


LENGTH    MOTION 


107 


and  that  the  length  of  the  arc  between  any  two  points  at  which 
X  =  a  and  x  =  b,  respectively,  is 


(7) 


'  =  J 


V«+(^)'-^ 


Vl  +ni-dx. 

The  equation  (5) 


62.   Motion  on  a  Curve.    Parameter  Forms 

of  §  Gl  is  often  written  in  the  form 
(8)  rfs2  =  fix2  _|.  ^y2, 

which  is  readily  remembered  through  the  suggestiveness  of  the 
triangles  3ILN  and  MLP  of  Fig.  25,  in 
which  ds  =  MP.  Equation  (8)  will  be 
called  the  Pythagorean  differential  for- 
mula. Since  m  =  tan  «  (Fig.  12,  p.  50), 
the  quantity  Vl  +  vi^  is  equal  to  sec  a. 
In  particular,  ds/dx  =  sec  a,  whence 
dx/ds  =  cos  a.  Likewise,  dy/ds  =  sin  a. 
If  a  point  3/  moves  along  a  curve,  its 
total  speed  v  is  ds/dt,  its  horizontal 
speed  v^  is  dx/dt ;  its  vertical  speed  v^ 
is  dy/dt 

(«)'      (IT=(IJHi)'""' 

the  square  of  the  total  speed  is  the  sum  of  the  squares  of  the  hori- 
zontal and  the  vertical  speeds;  and  u^=  v  cos  «,  v^  =  v  sin  a. 

The  equation  (8)  may  be  used  in  case  the  equations  of  the 
curve  are  given  in  parameter  form 

(10)  x=f{t),        y  =  <f>(t), 

whether  the  parameter  t  represents  the  time  or  some  other 
convenient  quantity.  The  length  of  an  arc  of  a  curve  whose 
equations  are  given  in  the  foira  (10)  is 


dx  =  Ax-^-^ 
Fig.  25. 
Dividing  both  sides  of  equation  (8)  by  (dty,  we  find 

2_,.  2_^^2_. 


(11) 


^=/v(f)^^(fr-^^- 


108  INTEGRATION  [V,  §  63 

63.  Illustrative  Examples. 

Example  1.     A  point  moves  along  a  curve  y  =  x"^.     Express  ds/dx  and 
write  the  integral  which  represents  the  length  of  the  curve. 

By  the  Pythagorean  formula  ds'^  =  dx^  +  dy^ ;   but  dy  =2xdx;  hence 


ds2  =  (1  +  4  a;2)  dx%   or   ^  =  VH-  4  x^. 
dx 

The  length  s  of  any  arc  between  points  where  x  =  a  and  a;  =  &  is 


']:::=jr 


V 1  +  4  x^  dx  : 


but  since  we  have  never  had  a  function  whose  derivative  is  v  1  +  4  x-, 
this  integral  cannot  now  be  found.  [See,  however,  Ex.  16,  p.  129,  and 
§  106.] 

The  speeds  v,  Vj„  and  Vy  are  given  by  the  relations  : 


dx 


v,  =  —  ,  'y„  =  ^  =  2x  —  =2  xv^,  V  =  VvJ  +  V  =  Vl  +  4  x^  v^. 
dt  dt  dt 

If  Vj.  is  given,  the  other  two  can  be  found  ;  thus,  if  Vj,  is  a  constant  k^ 

Vjc  =  k,   Vy  =  2  kx,   V  =  k  Vl  +  4  x^. 

Example  2.  Find  ds,  v,  s  for  the  curve  y"^  =  x^  and  find  the  length  of 
the  arc  from  the  origin  to  the  point  w^here  x  —  -5. 

Since  tj'^  =  x^,  we  have  2y  dy  =3  x-  dx,  or  4  y^  dy'^  =  9  x*  dx"^,  or  dy'^ 
=  f  X  dx2  ;  and  ds-  =  dx^  +  di/^  =  (1  +  |  x)  dx".  It  follows  that  the  speed 
of  a  moving  point  is 

and  that  the  length  of  the  arc  is 

s  =JV1  +  |X  dx  +  C=  5V(1  +  f  xf^  +  C, 

hence  the  length  between  the  origin  and  the  point  where  x  =  5  is 

]i=5  /»x=5      , 3/2-11=5 

=  \       Vl  +  |xdx  =  ^(l  +  |x)  =W 

Example  3.     Find  f7s,  v,  s  for  the  curve  represented  by  the  equations 

x  =  t^  +  5,    y  =  ^(4  t  +  ly/-. 
We  find  ds^  =  dx^  +  dy'^  =  (2 1  dVf  +  [(4  f  +  ly-l^-dtf  =  (2  «  -f  XydV^ ; 
whence  ds  =  (2  «  +  1)  dt,  or  v  =  ds/<^^  =  2  «  +  1,  and 

s=V^J-dt^  C={{2t^V)dl-\-  C^t-  +  t+  C. 


V,  §  63]  LENGTH    MOTION  109 

Between  the  points  where  t  =  0    and  where   t  =  2   [i.e.  the  points 
(a;  =  5;  y  =  1/6)  and  (x  =  9,  y  =  27/6)],  the  length  of  the  arc  is 
^-jr=2  ^  /.«=2      ^  _^  ^  ^^  =  r«2  +  f  +  c1'^^  =(4  +  2  +  C)  -  C  =  6. 

Our  present  ability  to  recognize  derivatives  enables  us  to  integrate  com- 
paratively few  of  the  square  root  forms  that  occur  in  these  length  integrals. 
We  shall  be  able  to  deal  with  these  forms  inore  readily  in  Chapter  "VI. 

EXERCISES   XXIII— LENGTH  — TOTAL  SPEED 

1.  Determine  by  integration  the  lengths  of  the  following  curves,  each 
between  the  limits  x=:ltox  =  2,  x  =  2tox  =  4,  x  =  rttox  =  6.  Check 
the  first  three  geometrically  : 

(a)y  =  2x-l.  (c)  ?/ =  TOX  +  c.  (e)     ?/ =  ^  (2  x  -  1)3/2. 

(6)y  =  3+4x.  (d)  2/  =  |(x-l)-'/2.        (/)  ^  =    i  (4.,,_  ])3/2. 

2.  Find  ds,  and  the  length  s  of  the  path  of  each  of  the  following  mo- 
tions, between  the  given  limits.     Find  the  speed  v  at  each  end  of  the  arc. 

(a)  X  =  1  +  t,  y  -1  -t;  t  =  0  tot  =  2. 

(b)  X  =  (1  +  ty^,  2/  =  (1  -  0^/- ;  ^  =:  0  to  «  =  1. 

(c)  X  =  (1  -  ty,  y  =  8  f^i"'!?, ;  <  =  0  to  «  =  0. 

(d)  x  =  \-\rt:-,y  -t  -  t;^/'i  ;  «  =  0  to  «  =6. 

(e)  X  =  2A,  y  =  t  +  1/(3  «3)  ;  t  ^  a  io  t  =  h. 

3.  If  a  point  moves  on  the  circle  x^  +  y- =  1,  show  that  x{dx/dt) 
+  yidy/dt)  =  0,  and  that  v^  =  [dx/rfiJV?/^  ^  [dy/dtY/x-^- 

4.  If  a  point  moves  on  the  circle  x^  +  ?/2  =  1  with  constant  speed  v  =  k, 
show  that  dx/dt  =  ±  ky  and  dy/dt  =  ±  kx,  where  the  sign  ±  depends  on 
the  sense  of  the  motion. 

5.  If  a  point  moves  on  the  hyperbola  xy  =  1,  show  that  the  horizontal 
and  the  vertical  speeds  v^  and  Vy  are  connected  by  the  relation  xuy+yt\=0  ; 
and  that  v^  =  Vr^(x'^+y^)/x^=Vy^(x^+y^)/y^. 

6.  If  a  point  moves  on  the  curve  y^  =  x,  show  that  v-=  (1  +  4  y-)Vi,'^. 

7.  Determine  the  path  described  when  the  x  and  y  speeds  are  as  be- 
low, if  the  point  is  at  (0,  0)  when  t  =  0.  Find  the  length  of  the  arc  trav- 
ersed from  t  =  0  to  t  =  9.     What  is  the  speed  at  each  end  of  these  arcs  ? 

[dt  [dt  [dt       ^^      '■ 


110  INTEGRATION  [V,  §  64 


PART   IT.     INTEGRALS   AS   LIMITS  OF   SUMS 

64.  Step-by  step  Process.  The  total  amount  of  a  variable 
quantity  whose  rate  of  change  (derivative)  is  given  [('.e.  the 
integral  of  the  rate]  can  be  obtained  in  another  way. 

For  example,  imagine  a  train  whose  speed  is  increasing. 
The  distance  it  travels  cannot  be  found  by  multiplying  the 
speed  by  the  time;  but  we  can  get  the  total  distance  approxi- 
mately by  steps,  computing  (approximately)  the  distance  trav- 
eled in  each  second  as  if  the  train  were  actually  going  at  a 
constant  speed  during  that  second,  and  adding  all  these  results 
to  form  a  total  distance  traveled. 

If  the  speed  increases  steadily  from  zero  to  30  mi.  per  hour,  in  44 
sec,  that  is,  from  zero  to  44  ft.  per  second  in  44  sec,  the  increase  in  speed 
each  second  (acceleration)  is  1  ft.  per  second.  Hence  the  speeds  at 
the  beginnings  of  each  of  the  seconds  are  0,  1,  2,  3,  •••  ,  etc. 

Using  the  speeds  as  approximately  correct  during  one  second  each, 
we  should  find  the  total  distance  (approximately) 

s  =  0  +  l  +  2  +  3  +  -+42  +  43=  ^A^  =  94(3, 

which  is  evidently  a  little  too  low. 

If  we  used  as  the  speed  during  each  second  the  speed  at  the  end  of 
that  second,  we  should  get  (approximately) 

s=l+2+3+4+ 

which  is  evidently  too  high.  But  these  values  differ  only  by  44  ft.  ; 
and  we  are  sure  that  the  desired  distance  is  between  94G  and  990  ft. 

If  we  reduce  the  length  of  the  intervals,  the  result  will  be  stiii  more 
accurate  ;  thus  if,  in  the  preceding  example,  the  distances  be  computed  by 
half  seconds,  it  is  easily  shown  that  the  distance  is  between  957  ft.  and 
979  ft. ;  if  the  steps  are  taken  1/10  second  each,  the  distance  is  found  to 
be  between  965.8  ft.  and  970.2  ft. 

Evidently,  the  exact  distance  is  the  limit  approached  by  this  step-by- 
step  summation  as  the  steps  At  approach  zero  : 


s  =(        vdt={ 

J(=0        Jt=o  Jt 


f=44  fl-y  '=44 

ff?«=l-  =968. 


V,  §  65] 


APPROXIMATE  SUMMATION 


111 


We  note  particularly  that  the  two  results  for  s  are  surely  equal ;  hence  we 
obtain  the  important  result : 


£ 


lim 


A«  +  r  •  A<  +  u  I  •  A(  +  ...  1. 

J  t=At  J  (=2A(  I 


65.  Approximate  Summation.  This  step-by-step  process  of 
sumiuatioii  to  find  a  given  total  is  of  such  general  application, 
and  is  so  valuable  even  in  cases  where  no  limit  is  taken,  that 
we  shall  stop  to  consider  a  few  examples,  in  which  the  methods 
employed  are  either  obvious  or  are  indicated  in  the  discussion 
of  the  example. 

Thus,  areas  are  often  computed  approximately  by  dividing  them  into 
convenient  strips.  We  have 
seen,  §  (50,  that  if  A  denotes 
the^  area  under  a  curve  be- 
tween X  =  a  and  x  =  b,  then 
the  rate  of  increase  of  A  is 
the  height  h  of  the  curve  : 


clA 

dz 


Ji(x), 


whei-e  Ii(x)    is   the   rate   of 

increase  of  A,  and  is  also  the  height  of  the  curve. 

For  a  parabola,  h  —  x^,  we  may  find  the  area  A  approximately  between 

X  —  —  1  and  X  =  2  by  dividing  that  interval  into  smaller  pieces  and  com- 
puting (approximately)  the  areas  which  stand 
on  those  pieces  as  if  the  height  h  were  con- 
stant throughout  each  piece.  If,  for  ex- 
ample, we  divide  the  area  A  into  six  strips  of 
equal  width,  each  1/2  unit  wide,  and  if  we 
take  the  height  throughout  each  one  to  be  the 
height  at  the  left-hand  corner,  the  total  area 
is  (approximately) 

(+l)2i  +  (|)H  =  li»/8, 
whereas,  if  we  take  the  height  equal  to  the 
Pjq   07  height  at  the  right-hand  corner  we  get  :il/8. 

The  area  is  really  3,  as  we  find  by  §  00.     Tak- 
ing still  smaller  pieces  the  result  is  of  course  better  ;  thus  with  30  pieces 


Xj-  5   it 

3  -3  "^/ 

ii^i^ii;ii 

--k'-f--- 

1                         1     i     1 

112  INTEGRATION  [V,  §  65 

each  jJjj  unit  wide,*  the  left-hand  heights  give  2.855,  the  right-hand  heights 
3.155.  With  still  more  numerous  (smaller)  pieces  these  approximate  re- 
sults approach  the  true  value  of  the  area.     (See  §  67,  p.  116.) 

EXERCISES  XXIV.    STEP-BY-STEP  SUMMATION  —  APPROXIMATE 
RESULTS 

1.  Approximate  to  about  1  %  the  areas  under  the  curves  below,  be- 
tween the  limits  indicated.  Estimate  the  answers  roughly  in  advance. 
Use  judgment  with  regard  to  scales  to  gain  in  accuracy  by  having  the 
figure  as  large  as  is  convenient.  Check  results  by  integration  where 
possible. 

(a)  2/  =  1  +  x2 ;  x  =  0  to  3.  (/)  2/  =  x-i ;  X  =  10  to  100. 

(6)  2/  =  ™  ;  X  =  5  to  10.  (9)  2/  =  (1  +  x)/^  )  a;  =  2  to  4. 

100  

(c)  2/  =  x2-2x;x  =  lto3.  (h)  y  =  V9  +  x  ;  x  =  0  to  7. 

(d)  2/  =  4x2  -  X*  ;  X  =  0  to  2.  (0   y  =  V9  +  x^ ;  x  =  0  to  4. 

(e)  2/  =  x-2  ;  X  =  1  to  10.  (j)   y  =  V9  +  x*  ;  x  =  0  to  2. 

2.  Approximate  to  about  1%  the  distance  passed  over  between  the 
indicated  time  limits,  where  the  speed  is  as  below ;  when  possible  check 
by  integration. 


(a)  v=l+  Vi;  t=Oto  100. 

(ft)  v=2t  +  t^;  «  =  1  to  4. 

(c)  V  =       ^       ■  t  —  0  to  100 

/  j\    „,  i           .    /  1    +/->  A 

1  + Vi' 

^'^^"-2i+e2'^-l*°4- 

(e)  v  =  '^;t  =  ltolO. 

(in^  =  f^f'  «=itoio. 

{g)  V  =  l±^  ;  «  =  4  to  9. 

Vt 

(A)  V  =      ^^     ;  i  =  4  to  9. 

1+  v< 

(i)   v  =  ^~^;  t  =  0to50. 

( j)  v=y/l  +  f^;  « =  10  to  20. 

3.  The  volume  of  a  metal  casting  is  often  found  by  dividing  the  entire 
pattern  into  parts,  each  of  which  can  be  computed  readily.  Show  how 
to  find  the  volume  of  a  flat  casting  shaped  like  the  letter  H,  if  the  thick- 
ness and  the  width  of  each  portion  is  given. 

*  In  this  computation  the  formula  12  +  22-1-32H h  n2=  n{2n+  1)(h+1)/6 

is  convenient.  For  any  reasonable  degree  of  accuracy,  this  method,  in  this 
example,  is  longer  than  that  of  §  (10,  but  fur  other  examples,  especially  when 
the  curve  is  drawn  and  we  know  no  equation  for  it,  this  method  is  often  con- 
venient. Notice  that  the  average  of  the  two  last  results  found  above  is 
reasonably  accurate  ;  it  is  3.005  ft.     (See  §  66,  p.  114.) 


V,  §65]  APPROXIMATE  SUMMATION  113 

4.  Show  how  to  calculate  approximately  the  volume  of  a  dumbbell 
whose  ends  are  spheres.  Notice  that  a  small  volume  at  the  intersection  of 
the  spheres  with  the  cross-bar  is  neglected. 

5.  Show  how  to  find  the  volume  of  a  cone  approximately,  by  adding 
together  layers  perpendicular  to  its  axis. 

6.  Find  the  volume  of  a  sphere  by  imagining  it  divided  into  small 
pyramids  with  their  vertices  at  the  center  and  their  bases  in  the  surface, 
as  in  elementary  geometry. 

7.  Discuss  the  approximate  evaluation  of  areas  in  a  plane  by  counting 
the  squares  in  a  figure  drawn  on  cross-section  paper.  Would  still  more 
finely  ruled  paper  be  more  accurate  ?  Show  that  the  area  of  any  closed 
figure  may  be  defined  by  extending  this  process  indefinitely. 

8.  The  volume  of  a  ship  is  computed  by  means  of  the  areas  of  cross 
sections  at  small  distances  from  each  other  ;  show  how  the  result  is  cal- 
culated. Show  how  to  make  a  more  accurate  computation  by  the  same 
method. 

9.  In  shipments  of  ores  or  coal,  it  is  usual  to  sample  each  car  ;  show 
how  to  obtain  the  total  amount  of  metal  in  a  shipment  of  several  car-loads 
of  ore.  Is  the  result  accurate  or  approximate  ?  Show  how  a  more  accu- 
rate result  can  be  found. 

10.  The  number  of  bacteria  in  a  river  is  computed  by  sampling  at 
various  distances  from  the  shore.  Show  that  the  total  thus  computed  is 
reasonably  accurate,  on  the  assumption  that  the  bacteria  per  cubic  foot 
is  approximately  constant  for  short  distances. 

11.  The  total  sales  of  a  given  stock  or  bond  in  one  year  on  the  New 
York  Stock  Exchange  can  be  computed  from  the  record  of  the  number 
sold  each  day  and  the  price  on  that  day.  Show  that  the  result  lies  be- 
tween that  found  by  using  the  highest  and  the  lowest  daily  prices.  Would 
the  average  of  the  latter  two  be  more  accurate  ? 

12.  The  number  in  100,000  persons  alive  at  any  given  age  who  die  be- 
fore they  are  one  year  older  is  important  in  life  insurance  ;  show  how  to 
compare  the  actual  death  rate  of  a  given  group  of  people, — say  of 
the  students  in  a  given  university,  —  with  the  published  figures  showing 
the  normal  expectation  of  death  during  each  year  of  age. 

13.  The  amount  of  cement  used  in  concrete  varies  in  different  portions 
of  the  same  building  from  one  part  in  two  to  one  part  in  six.  Show 
how  to  find  the  entire  amount  of  cement  used  in  the  work  from  the 
specifications. 


114  INTEGRATION  [V,  §  66 

66.  Exact  Results.  Summation  Formula.  As  in  the  preced- 
ing articles,  given  the  rate  of  increase  ]i(x)  of  a  variable 
quantity  y  we  can  always  compute  the  total  difference  in 
the  values  of  y  between  two  values  of  x,  x  —  a  and  a;  =  6, 
[ i. e.   the  integral ilzlR(x) dx^  : 

Let  us  break  up  the  interval  x  =  a  to  x  =  b  into  n  portions, 
each  of  size  Ax ;  the  first  interval  is  from  a  to  a  +  Ax,  the 
second  from  a-{-  Ax  to  a  +  2  Ax,  and  so  on.  The  change  in  y 
during  each  interval  can  be  computed  approximately  by  taking 
the  rate  of  change  as  constant  and  equal  to  its  value  at  the  be- 
ginning of  the  interval ;  doing  so  we  would  obtain,  in  the  first 
interval  a  change  Ax  •  E(a);  in  the  second  Ax  •  E(a-\-  Ax)  ;  in 
the  third  Ax  •  E  (a -{- 2  Ax) ;  etc.,  so  that  the  total  change  is 
(approximately)  the  sum  : 

(1)  s  =  AX'  E  (a)  +  Ax-  E(a  +  Ax)  -{-Ax-  E(a  +  2  Ax)  + 

•••  +AX'  E[a  +  (71  -  2)  Ax^  +  Ax  ■  E[a  + (n~l)Ax']. 

If  we  take  the  constant  rate  as  the  rate  at  the  end  of  the 
interval,  we  get  the  sum 

(2)  S  =  Ax-  E(a  +  Ax)  +  Ax-E (a +  2  Ax)  +  AxE (a -|- 3  Ax) -f 

\-Ax-E[^a  +  (n  —  1)  Ax]  -|-  Ax-E  (a  +  n  •  Ax). 

The  first  of  these  sums  contains  the  term  Ax  •  E  (a),  the 
second  the  term  Ax  •  72  (a  -}-  w  •  Ax)  ;  their  difference  is 
D  =  S  -s  =  Ax  .  [E(a-{-7i  •  Ax)  -  i?  (a)]  =  Ax  [72  (6) -7?  (a)] 
since  b  =  a-\-7i  •  Ax.  If,  for  example,  the  rate  E(x)  is  increas- 
ing, the  correct  answer  evidently  lies  between  S  and  s.  S  is 
too  high,  s  is  too  low.  As  we  make  the  intervals  smaller  and 
more  numerous,  Ax  will  approach  zero,  n  Avill  become  infinite,* 
and  D  =  S  —  s  =  Ax[E(b)  —  E(a)^  will  approach  zero,  since 
E(b)  —  E{a)  is  a  constant. 

Hence  it  is  evident  that  the  correct  value  of  the  total  change  iti 
y  is  the  li7nit  of  the  simi  s  (or  of  the  sum  S,  since  the  difference 
between  S  and  s  approaches  zero)  ;  that  is: 

*  For  the  meaniug  of  this  phrase,  see  §  14,  p.  19. 


V,  §  66] 


EXACT  SUMMATION 


115 


(^)I 


R  (x)  dx 


lim    \i^x  ■  R  (rt)  +  Aa^  ■  -B  (ffi  +  Ax)  + 

Aa>=0  / 


+  A.r  ■ltia+  (n  -  1)  Aj-] 


This  foniuila  will  be  called  the  Summation  Formula  of  the 
Integral  Calculus. 

Interpreted  as  a  motion  problem^  B  (x)  means  the  speed,  x  denotes 
time,  y  distance  ;  the  intervals  A.c  ai'e  small  intervals  of  time  during 
which  we  conceive  the  speed  as  sensibly  constant ;  Ax  •  R  (a)  is  the  dis- 
tance (approximately)  traversed  in  the  first  interval,  during  which  the 
speed  is  supposed  to  remain  approximately  equal  to  the  speed  S  (a)  at 
the  beginning  of  the  interval  ;  and  so  on,  as  in  the  example  of  §  65. 

Graphically,  if  x  and  y  denote  any  concrete  quantities  one  pleases, 
drawing  z  horizontally  as  usual,  we  may  represent  the  rate  R{:i-)  by  a 


h=Ii{x) 


¥ui.  28. 


curve  whose  height  is  7i :  h  =  B  (x).  The  intervals  Ax  are  small  intervals 
along  the  .r-axis  ;  PPu  PiP-z,  P^Ps,  •••,  in  each  of  which  we  think  of  the 
height  h  =  B(x)  as  sensibly  constant.  The  product  Ax  •  B  (a)  is  the  area 
7'/'iA'iJ/of  the  rectangle  whose  base  is  Ax  and  whose  height  is  PM  = 
h]^^a  =  Bia).  The  next  term  of  the  sum  s  is  Ax  ■  Bia  +  Ax),  which 
is  the  area  of  the  rectangle  P\P.K>L\,  and  so  on  ;  the  whole  sum  s  is  the 
area  of  the  polygon  PQKsLiJuLsh'iLiKzLih'iM  in  Fig.  28,  in  which  7i  is 
taken  equal  to  5. 


116 


INTEGRATION 


[V,  §66 


Likewise  the  sum  ^S*  is  the  area  (i'ig.  29)  of  the  polygon  PQNJ^LtJi 
L3J3L2J2L1J1,  which  is  exterior  to  the  curve. 

.7,      IZ^^^       h=R{x) 
h 


lim  S-- 

Ax=0 


lim  s 


lim  {Ax  B{a)  + 


+^xB\_a-\-{n—\)^x]} 


Fig.  29. 


The  difference  D  —  8  —  s  :=  6.x[^B  {h)  —  B{a)']  is  the  area  of  a  rec- 
tangle whose  base  is  Ax  and  whose  altitude  [_B{h)  —  B{a)'\  is  the  differ- 
ence between  PM  and  QN ;  and  it  is  evident  that  this  area  approaches 
zero  with  Ax. 

The  area  of  either  polygon,  s  or  *S',  evidently  approaches  the  area  A 
under  the  curve  between  x  =  rt  and  x  =  6  as  Ax  approaches  zero  : 

(4)  ^]^ 

which  agrees  with  our  previous  formulas  since 

=  \       hdx=  \        B(x)dx, 

and  the  two  values  of  A  agree,  by  (3).  This  agreement  maybe  regarded, 
however,  as  a  new  proof  of  (3),  since  the  two  formulas  for^  are  obtained 
independently  ;  but  attention  is  called  to,^  the  fact  that  this  argument  is 
simply  a  special  case  of  the  general  argument  used  above. 

It  is  evident  from  the  figures  that  (3)  holds  also  if  R(x)  is 
decreasing,  or  indeed  even  if  R(x)  changes  from  increasing  to 
decreasing,  or  conversely. 


67.  Integrals  as  Limits  of  Sums.  By  far  the  greater  number 
of  integrations  appear  more  naturally  as  limits  of  su7ns  than  as 
reversed  rates. 


v,§ 


EXACT  SUMMATION 


11? 


Thus,  as  a  matter  of  fact,  even  the  area  A  under  a  curve, 
treated  in  §  60  as  a  reversed  rate,  probably  appears  more 
naturally  as  the  limit  of  a  sum,  as  in  (4),  §  66.  Of  course  the 
two  are  equivalent,  since  (3),  §  66,  is  true ;  in  any  case  the  re- 
sults are  calculated  always  either  approximately,  as  in  the 
exercises  under  §  65,  or  else  precisely  by  the  methods  of  §§  58- 
59.  Hence  the  method  of  §  59  was  given  first,  because  it  is 
used  for  each  calculation  even  when  the  problem  arises  by  a 
summation  process. 

On  account  of  the  frequent  occurrence  of  the  summation 
process,  we  may  say  that  an  integral  really  means*  a  limit  of  a 
sum,  but  when  absolutely  precise  results  are  wanted  it  is  calcu- 
lated as  a  reversed  dijferentiation.'f  The  symbol  J  is  really  a 
large  S  somewhat  conventionalized,  while  the  dx  of  the  symbol 
is  to  remind  us  of  the  Ax  which  occurs  in  the  step-by-step 
summation. 

68.  Water  Pressure.  As  another  typical  instance,  consider 
the  water  pressure  on  a  dam  or  on  any  container. 

The  pressure  in  water  increases 
directly  with  the  depth  s,  and  is  equal 
in  all  directions  at  any  point.  The 
pressure  p  on  unit  area  is 

(1)  p^k.s 

where  s  is  the  depth  and  k  is  the 
weight  per  cubic  unit  (about  62.4  lb. 
per  cubic  foot).  I,,     .;,, 

Suppose  water  flowing  in  a  para- 
bolic channel   (Fig.   30),  the  parabola  being  defined  by  the  equation 


1 

1          1            100    !      1 

1 

' 

1  i 

II       Ml 

i-UJ 

I        1            ■        '    ' 

/■ 

Tvo 

--\ 

/  . 

" 

*  It  is  really  a  waste  of  time  to  discuss  at  great  length  here  which  fact 
about  integrals  is  used  as  a  definition,  and  which  one  is  proved  ;  to  satisfy  the 
demand  for  formal  definition  the  integral  may  be  defined  in  either  way,  —  as 
a  limit  of  a  sum,  or  as  a  reversed  differentiation.  The  important  fact  is  that 
the  two  ideas  coincide,  which  is  the  fact  stated  in  the  Summation  Formula. 

t  Later  we  return  to  approximate  methods  of  calculation.     (See  §  125.) 


Ah 


118  INTEGRATION  [V,  §  68 

(2)  10-^  =  225  h, 

where  to  is  the  width  and  h  is  the  height  above  the  bottom.  J^et  a  be  the 
total  depth  of  water  in  the  channel.  Then  the  depth  s  at  any  point  is 
s  =  a—  h,  and  the  pressure  is 

(3)  p  =  k(a-h). 

If  the  water  is  stopped  by  a  cut-off  gate,  the  total  pressure  on  the  gate 
is  most  easily  computed  by  dividing  the  gate  into  horizontal  strips  of 
height  Ah  each ;  throughout  one  of  the  strips  the  pressure  is  very  nearly 
constant  ;  the  total  pressure  on  a  strip  is  (approximately)  the  product  ol 
its  area  and  the  pressure  per  unit  area : 

(4)  pressure  on  each  strip  =  {^o  •  Ah]  •  {p}  =  p  -w  •  Ah, 
so  that  the  total  pressure  P  on  the  gate  is  (approximately) 

(5)  P  =     fp  .  tol         .  A/i  +  fn  .  Mj]  ■  Ah+  •■■  +\p-  w] 

[L  Jh=Ah  L  Ja  =  2AA  L  jA=n. 

where  n  is  the  number  of  strips.     The  exact  value  is  therefore 

(6)  P]  """  =  lim   \  [inv']         ■  Ah  +  [ pio] 

+    pio\  ■  Ah\  =  \     .  picdh, 

^y  (•^))  P-  115.  In  the  problem  before  us,  w  —  15  /i^/-  and  p  =  k(a  —  h)  ; 
hence 

=   \        Vo  k{a  -  h)hy^  dh  =  15  A-  1^  -  '4-1        =  -i  ^•«"'. 

that  is,  the  total  pressure  P  on  the  gate  increases  as  the  fifth  power  of  the 
square  root  of  the  total  depth  a  of  water  in  the  channel ;  e.g.  four  times  the 
depth  of  water  would  mean  32  times  the  pressure. 

Note  that  the  formulas  (3),  (4),  (5),  (6)  apply  in  any  similar  example. 

It  is  important  to  notice  that  the  total  pressure  up  to  any  height  h  =  h 
is  a  function  of  h  whose  rate  of  change  is  p-w.  Thus,  if  the  gate  be 
made  in  two  parts,  the  lower  portion,  of  height  7t,  bears  a  pressure 

P,.pT='^=304^-^r^:.30A:(^-^^-^% 
J;,=o  L     3  5  J;,=o  \    3  5   y 

The  rate  of  change  of  P,,  as  h  increases  is  dP,Jdh  =  l^i  k{n—h)h'^^-  —p-w. 

In  general,  if  the  height  of  the  lower  portion  of  the  gate  be  incrt'ased 

by  an  amount  Ah,  the  pressure  P^  on  the  portion  is  increased  by  an  amount 


■Ah 


V,  §68]  EXACT  SUMMATION  119 

^P^  =  p  ^nAh,  approximately,  so  that  A P^/ Ah  =  p  ■  to  (nearly)  ami 
(ir,,/dh  =  i^  .w  where  p  is  the  pressure  at  the  upper  edge  of  the  lower  por- 
tion. The  integral  in  (G)  may  be  thought  of  as  the  reversal  of  this  rate, 
as  ill  §§  64,  ()6. 

This  argument  is,  however,  by  no  means  so  natural  as  the  above  argu- 
ment by  summation.  The  important  thing  to  notice  is  that  even  in  this 
case  the  integrated  function  is  really  the  rate  of  increase  of  P  as  a  func- 
tion of  h.  But  in  some  problems  it  is  difficult  to  show  directly  that  the 
integral  is  a  reversed  rate,  except  by  using  (3).  The  great  value  of  the 
summation  formula  (.•>),  §  66,  is  that  it  makes  it  unnecessary  for  us  to 
express  each  problem  as  a  reversed  rate. 

EXERCISES   XXV.  — INTEGRALS   AS  LIMITS  OF  SUMS 

Determine  the  following  quantities,  (a)  approximately  by  step-by-step 
summation  ;   (h)  exactly  by  integration  between  limits  : 

1.  The  area  under  the  curve  ?/  =  z-  from  x  =  1  to  x  =  3  ;  from  x  =  a 
to  X  =  6. 

2.  The  area  under  the  curve  y  =  x^  from  x  =  0  to  x  =  2  ;  from 
X  =—  1  to  X  =-f  1. 

3.  The  area  under  the  curve  x-y  =  1  from  x  =  2  to  x  =  5. 

4.  The  distance  passed  over  by  a  body  whose  speed  is  u  =  2 «  -|-  10 
from  «  =  0  to  f  =  3. 

5.  The  distance  passed  over  by  a  falling  body  (v  =  gt)  from  <  =  2  to 
t  =  5. 

6.  The  increase  in  speed  of  a  falling  body  from  the  fact  that  the 
acceleration  is  ^  =  32.2,  from  «  =  0  to  «  =  3. 

7.  The  increase  in  the  speed  of  a  train  which  moves  so  that  its  accel- 
eration is  j  =  </100,  between  the  times  t  =  0  and  t.  =  3.  The  distance 
passed  over  by  the  same  train,  starting  from  rest,  during  the  same  interval 
of  time. 

8.  The  number  of  revolutions  made  in  5  min.  by  a  wheel  which 
moves  with  an  angular  speed  w  —  ^-/lOOO  (radians  per  second). 

9.  The  time  required  by  the  wheel  of  Ex.  8  to  make  the  first  ten 
revolutions. 

10.  Repeat  Ex.  8  for  a  wheel  for  which  a  =  100—  10 1  (degrees  per 
second).  Find  the  time  required  for  the  first  revolution  after  t  =  0  ; 
note  that  the  speed  is  decreasing 


120  INTEGRATION  [V,  §  68 

11.  Find  the  total  pressure  in  tons  on  one  side  of  the  gate  of  a  dry- 
dock,  the  wet  area  of  the  gate  being  a  rectangle  80  ft.  long  and  30  ft. 
deep. 

12.  The  pressure  in  pounds  on  one  side  of  a  board  10  ft.  long  and  2  ft. 
wide,  which  is  submerged  vertically  in  water  with  the  upper  end  10  ft. 
below  the  surface. 

13.  The  pressure  on  an  equilateral  triangle  20  ft.  on  a  side,  submerged 
in  water  with  its  plane  vertical  and  one  side  in  the  surface. 

14.  The  pressure  on  one  side  of  a  square  tank  10  ft.  high  and  5  ft.  on 
a  side,  the  tank  being  filled  with  a  liquid  of  specific  gravity  .8. 

15.  The  pressure  on  one  face  of  a  square  10  ft.  on  a  side,  submerged 
so  that  one  diagonal  is  vertical  and  one  corner  in  the  surface. 

16.  The  pressure  on  one  end  of  a  parabolic  trough  filled  with  water, 
the  depth  being  3  ft.  and  the  width  across  the  top  4  ft. 

17.  The  pressure  to  1%  on  a  circular  disk  10  ft.  in  diameter,  sub- 
merged below  water  with  its  plane  vertical  and  its  center  10  ft.  below  the 
surface. 

18.  The  weight  of  a  vertical  column  of  air  1  ft.  in  cross  section  and 
1  mi.  high,  given  that  the  weight  of  air  per  cubic  foot  at  a  height  of  h  feet 
is  .0805  -  .00000268  h  pounds. 

69.   Volumes.  —  The  volumes  of  many  solids  may  be  com- 
puted readily  by  the  summation  process,  either  approximately, 
as  in  §  65,  or  exactly  by  using  the  Summation  Formula,  which 
leads  to  a  reversal  of  a  differentiation.     We  proceed  to  illus- 
y  trate  this  application  by  examples. 

/    \  Exainple.    To   find   the  volume  of  a 

Py^"  T'"~-vV  0  right  circular  cone  whose  height  k  is  10 

/S^?/mf^\\^^m\  ft-  ^^^  thte  radius  of  whose  base  a  =  4  ft. 

/                         \  Let  s  be  the  distance  from  the  ver- 

/  .— -_\.  tex  F  to  any  plane  PQ  parallel  to  the 

/j/^                o\                    \^  base  of  the  cone.    The  section  of  the 

^ /_ cone  by  this  plane   is  a  circle  whose 

/  radius  r  =  CQ  is  as/h,   since  the  tri- 
angles   VOE   and    VCQ    are    similar; 


Fig.  31. 


hence  the  area  As  of  this  circular  section  is  : 

(1)  As  =  TTV  =  — ^ 


V,  §70]  VOLUMES  OF  SOLIDS  121 

If  we  divide  up  the  whole  solid  into  layers  of  thickness  As,  the  volume 
of  each  layer  is,  approximately,  the  product  of  its  thickness  As  times  the 
area  As  of  the  bottom  of  the  layer  : 

(2)  Volume  of  one  layer  —  Vs  =  AsAs; 

since  the  value  of  s  at  the  bottom  of  the  first  layer  is  As,  the  value  of 
s  at  the  bottom  of  the  second  layer  is  2  As,  etc.,  the  total  volume  is, 
approximately, 

(3)  As]         -As+As]  -As  +  .-.  +  ^sl  As, 

J  s=A»  J  s=2  Aa  J  «=n  •  As 

where  71  is  the  number  of  layers.     Therefore  the  total  volume  is 

(4)  Fl'"*  =  lim  1^5]         -As  +  ^sl  -As+.-.+^sl  -As) 

J«=0        As=y)  [         Js=A8  J«=2As  J»=nA.  J 

rs=h 

=  \       Asds, 

Js=0 

by  the  Summation  Formula  (3),  p.  115.  Substituting  the  value  of  As 
from  (1),  we  find  : 

(5)  f]^=  r*^sczs=  pz^rf,^  r-a^sfy-^^ 

J,=o     J.=o      ^         1=0     h-  Ih^  3]s=o        3    ' 

which  agrees  with  the  formula  of  ordinary  geometry.  In  this  problem, 
the  given  values  are  A  =  10,  a  =  4  ;  hence,  we  find  T'=1G7.5  cu.  ft. 
(nearly). 

Notice  that  it  is  quite  true  that  the  rate  of  increase  of  Fas  a  function 
of  s  is  wa^s^/h^  ;  in  general  an  increase  in  height  As  causes  an  increase  in 
volume  As  ■  As  (nearly)  where  As  is  the  area  of  the  bottom  of  the  layer 
added  ;  hence  dV/ds  =  As,  which  agrees  with  the  integral  formula  (4), 
as  in  §§  66-68. 

70.  Volume  of  Any  Frustum.  A  solid  which  is  bounded  at 
two  extremities  by  a  pair  of  ])arallel  planes  is  called,  in  gen- 
eral, a  frustum.* 

If  such  a  frustum  be  divided  up  into  layers  of  thickness  As, 
as  in  §  69,  by  planes  parallel  to  the  base,  and  if  ^1^  represents 
the  area  of  any  section  at  a  distance  .s-  from  the  upper  bounding 

*  In  special  cases,  a  frustum  may  touch  one  or  botli  of  the  bouiidii)<c  paral- 
lel planers  in  a  single  point;  such  special  cases  include,  for  example,  the 
sphere ;  see  Ex.  1  below.  The  two  parallel  planes  which  bound  the  solid  are 
called  truncating  planes. 


122 


INTEGRATION 


[V,  §  70 


plane,*  the  formulas  of  §  69  numbered  (2),  (3),  (4)  all  hold,  the 
arguments  being  unchanged.  If  the  area  As  is  known  in  terms 
of  s,  say  As  =/(s),  (4)  becomes 

this  formula  will  be  called  the  Frustum  Formula.  It  may  be 
used  to  find  the  volume  of  any  solid,  if  we  know  how  to  find 
the  areas  of  any  complete  set  of  parallel  cross  sections. 


^ 

I 
\ 

^ 

1 

fes 

a 

.6 

-'  ^ 

>*' 

a 

_-H*'^ 

\ 

Fig 

} 

/ 

/  ^s^ 

1          r 

.  33. 

1 

/ 

F 

G.  32. 

Example  1.     To  find  the  volume  of  a  sphere  of  radius  a. 

The  sphere  may  be  thought  of  as  located  between  two  parallel  tangent 
planes  at  a  distance  2  a  from  each  other.  A  section  parallel  to  one  of  the 
planes  is  a  circle  of  radius  r  =  CN ;  its  ayea  is 

but,  in  the  triangle  OCN, 

OC-a-s,  0N=  a, 
whence  r^  =  CN^^  ON^  -  OC^  =^  a^  -(a  -  sy  =  2  as  -  s^ 

It  follows  that 


In  any  case,  s  may  be  counted  from  the  lower  bounding  plane,  if  conven- 


ient. 


V,  §  70] 


VOLUMES  OF  SOLIDS 


123 


-  <t=2,i  rs=ia 

Js=0  »'s=0 


A,.ds 


(2  as  —  s^)  (Is  ■■ 


L  3j.^  3 


This  is,  of  course,  the  usual  formula ;  notice  that  the  volume  of  a  hemi- 
sphere results  by  taking  the  limits  s  =  0,  and  s  —  a,  or  also  by  taking 
s  =  a  and  s  =  2a.  The  volume  of  any  frustum  of  a  sphere  may  be  ob- 
tained by  substituting  the  correct  values  of  s  for  the  limits  of  integration  ; 
thus  the  portion  of  a  sphere  cut  off  by  a  plane  at  a  distance  of  a/2  from 
the  center  is 


as  —  s^)  (Is 


■    as- 


^r 


3j,=o        24^'"      32I3' 


that  is,  5/32  of  the  volume  of  the  whole  sphere. 

Example  2.     To  find  the  volume  of  the  solid  foi-med  by  revolving  the 
curve  y  =x^  about  the  y-axis  ;  from  the  vertex  to 
the  point  where  y  =  4. 

The  solid  described  is  contained  between  the 
parallel  planes  y  =  0  and  y  =  4.  The  section  As  at 
any  height  A  is  a  circle  ;  its  area  is 

vis  =  i"?'^, 

where  r  is  the  radius  of  the  section.  But  h  and  r 
stand  for  values  of  y  and  x,  respectively ;  hence 
h  =  f^  and 

^5  =:  irx~  =  wy. 
Applying  the  frustum  formula,  we  have 

=  i        TTX^dy-  \       -rrydy- 

In  general,  the  volume  formed  by  revolving  any  curve  y  =f{x) 
about  the  i/-axis  between  two  planes  at  heights  a  and  b  is 


2  J,=o^ 


Stt. 


Jj/=a       Jy=a 


where  x^  must  be  replaced  by  its  value  in  terms  of  y  from  the  equation  of 
the  curve. 

Similarly  the  volume  of  a  solid  of  revolution  formed  by  revolving 
the  curve  y  =  /(.r)  about  the  u-'-axis  between  the  planes  x  =  a  and  x  =  b 


jx=a       Jx-a  Jx=a 


124  INTEGRATION  [V,  §  70 

EXERCISES   XXVI.— VOLUMES  OP   SOLIDS.     FRUSTA 

1.  Find  the  volume  of  a  frustum  of  a  cone  of  height  h,  if  the  radii  of 
the  two  bases  are,  respectively,  a  and  b. 

2.  Find  the  volume  of  the  paraboloid  of  revolution  formed  by  revolv- 
ing y-  —  4x  about  the  x-axis,  between  x  =  0  and  x  =  i;  between  a;  =  1 
and  X  =  5 ;  between  x  =  a  and  x  =  b. 

3.  Find  the  volume  of  a  hemisphere,  using  layers  parallel  to  each  other. 

4.  Find  the  volume  of  the  ellipsoid  of  revolution  formed  by  revolving 
an  ellipse  (1)  about  its  major  axis  ;  (2)  about  its  minor  axis. 

5.  Find  the  volume  of  the  portion  of  the  hyperboloid  of  revolution 
formed  by  revolving  about  the  ^/-axis  the  portion  of  the  hyperbola 
x2  _  j/2  _  1  between  y  =  0  and  y  =  2. 

6.  Find  the  volume  of  the  portion  of  the  hyperboloid  of  revolution 
formed  by  revolving  x^  —  y^  =  l  about  the  x-axis,  between  x  =  1  and  x  =  3. 

7.  Find  the  volumes  formed  by  revolving  each  of  the  following  curves 
about  the  x-axis,  between  x  =  0  to  x  =  2  ;  between  x=— ltox=:-|-l: 

(a)  y  =  x^.  (c)  y  =  x^  —  X.  (e)    y^  =  x  +  2y. 

(b)  2/  =  x2  -  1.  (d)  2/  =  (1  -I-  x)2.  (/)  V.,rn  +  Vy  =  4. 

8.  Proceed  as  in  Ex.  7  for  each  of  the  following  curves,  between 
sc  =  1  and  X  =  3  ;  between  x  =  a  and  x  =  b  : 

(a)2/  =  ii^.  (b)xy  =  l+x^.  (c)  x^  -  xV  =  1- 

9.  Proceed  as  in  Ex.  7  for  each  of  the  following  curves,  revolved, 
however,  about  the  y-axis,  between  y  =  0  and  y  =  2: 

(a)  x  =  2/3.  (c)   X  =  4  2/2  _  yS,  (e)    ^r.  -  y"- -  y. 

(b)  x2  =  yK  (d)  x'^  +  y*  =  81.  (/)  x  =  y^/^  +  yV*. 

10.  Find  the  volume  generated  when  the  segment  of  a  parabola  from 
its  vertex  to  its  focus  revolves  (1)  about  the  tangent  at  the  vertex; 
(2)  about  the  latus  rectum. 

11.  Find  the  volume  generated  when  the  area  between  the  parabola 
2/  =  6  X  —  x^  and  the  x-axis  revolves  about  the  x-axis. 

12.  Find  the  volume  generated  when  the  area  bounded  by  the  curves 
2/  =  x2  and  2/^  =  x  revolves  about  the  x-axis. 

13.  Calculate  the  volume  of  a  parabolic  trough  10  ft.  long,  3  ft.  deep, 
and  4  ft.  wide  at  the  top. 


V,  §71]  PRISMOID  FORMULA  125 

14.  Find  the  volume  generated  by  a  square  of  variable  size  perpen- 
dicular to  the  a;-axis,  which  moves  from  x  =  0  to  a;  =  5,  if  the  length  of 
the  side  of  the  square  is  (1)  proportional  to  x  ;   (2)  equal  to  y:^. 

15.  Find  the  volume  generated  by  a  variable  equilateral  triangle  per- 
pendicular to  the  X-axis,  which  moves  from  x  =  0  to  x  =  2,  if  a  side  of 
the  triangle  is  (1)  equal  to  X'^ ;  (2)  proportional  to  2  —  x. 

16.  Find  the  volume  generated  by  a  variable  circle  which  moves  in  a 
direction  perpendicular  to  its  own  plane  through  a  distance  10,  if  the 
radius  varies  as  the  cube  of  the  distance  from  the  original  position. 

17.  Find  the  mass  of  a  right  circular  cylinder  of  variable  density,  if 
the  density  varies  (1)  directly  as  the  distance  from  the  base;  (2)  in- 
versely as  the  square  root  of  the  distance  from  the  base. 

71.   Cavalieri's  Theorem.     The  Prismoid  Formula.     If  two 

solids  contained  between  the  same  two  parallel  planes  have  all 
their  corresponding  sections  parallel  to  these  planes  equal,  i.e. 
if  the  area  A' s  of  such  a  section  for  the  first  solid  is  the  same  as 
the  area  A" g  of  the  second,  it  follows  from  §  70  that  their  total 
volumes  are  equal,  since  the  two  volumes  are  given  by  the  same 
integral. 

This  fact,  known  as  Cavalieri's  Theorem,  is  often  useful  in 
finding  the  volumes -of  solids. 

If  the  area  As  of  any  section  of  a  frustum  is  a  quadratic 
function  of  s :  * 

(1)  As  =  as-  -f-  6s  -f  c 

■where,  as  in  §  70,  s  represents  the  distance  of  the  section  ^4^ 
from  one  of  the  two  parallel  truncating  planes,  the  volume  is 


(2) 


~\s=h  /^s=h  r       .,3  ^2  ~\s=h 

V\      =J_^    (as^  +  Ss-f  c)rfs=    a|  +  6|  +  cs  r 


where  h  is  the  total  height  of  the  frustum. 

*  It  is  shown  in  Ex.  3,  p.  128.  that  the  results  of  this  section  hold  also  when 
As  is  any  cubic  function  of  s :  As=  as^  +  bs'2  +  cs  +  d.  Notice  also  that  any 
linear  function  65  +  c  is  a  special  case  of  (1),  for  a  =  0. 


126  INTEGRATION  [V,  §  71 

The  area  B  of  the  base  of  the  frustum,  the  area  T  of  the  top, 
and  the  area  M  of  a  section  midway  between  the  top  and 
bottom  are 

B  =  .t n        =  [as'  +  hs  +  c1        =c  ; 

T==  As\        =\  as-  +  bs  4-  c  |        =  air  +  hh  +  c ; 

3f  =  .1 J        =  [as'  +  6s  +  c~|        =  a  J  +  &  .^'  +  c. 

If  we  take  the  average  of  B,  T,  and  4  times  Jf : 

B+  T+^M  ^alr      hh 

6  3         2' 

this  average  section  multiplied  by  the  total  height  h  turns  out 
to  be  exactly  the  entire  volume : 

(3)  B+T  +  ^M  ^  ^  ^  2|!  +f V  ch  =  rl:'\ 

This  fact  is  known  as  the  Prismoid  Formula.  It  is  easy  to  see  by 
actually  checking  through  the  various  fornuilas,  that  this  formula  holds 
for  every  solid  lohose  volume  is  given  in  elementary  geometry;  the 
same  formula  holds  for  a  great  variety  of  other  solids.*  But  the  cliief 
use  to  which  the  formula  is  put  is  for  practical  approximate  computation 
of  volumes  of  objects  in  nature  :  it  is  reasonably  certain  that  any  hill,  for 
example,  can  be  approximated  to  ratlier  closely  either  by  a  frustum  of  a 
cone,  or  of  a  sphere,  or  of  a  cylinder,  or  of  a  pyramid,  or  of  a  paraboloid  ; 
since  the  prismoid  formula  holds  for  all  these  frusta,  it  is  quite  safe  to 

*  The  formula  holds  also,  for  example,  for  any  }m)^moid,  i.e.  for  a  solid  with^ 
any  base  and  top  sections  whatever,  with  sides  formed  by  straight  lines  join- 
ing points  of  the  base  to  points  of  the  top  section.  For  example,  any  wedge, 
even  if  the  base  be  a  polygon  or  a  curve,  is  a  prismoid.  The  solids  defined  by 
(1)  include  all  these  and  many  others;  for  example,  spheres  and  paraboloids, 
which  are  not  prismoids.  The  formula  holds  for  all  these  solids  and  even 
(see  Ex.  3,  p.  128)  for  all  cases  where  As  is  any  cubic  function  of  s.  One 
advantage  of  the  formula  is  that  it  is  easy  to  remember:  even  the  formula 
for  the  volume  of  a  sphere  is  most  readily  remembered  by  remembering  that 
the  prismoid  formula  holds. 


V,  §  71] 


PRISMOID   FORMULA 


127 


use  the  formula  icithuut  even  truubluu/  to  see  ivhich  of  these  solids  actually 
approximates  to  the  hill.  Similar  remarks  apply  to  many  other  solids, 
such  as  metal  castings,  though  it  may  be  necessary  to  use  the  formula 
several  times  on  separate  portions  of  such  a  complicated  object  as  the 
pedestal  of  a  statue,  or  a  large  bell  with  attached  support  and  tongue. 

Example  1.     In  the  frustum  of  a  paraboloid  computed  in  Ex.  2,  p.  123, 
it  is  only  necessary  to  notice  that  the  formula  for  any  section 

As  =  Try 
is  a  quadratic  function  of  the  distance  y  from  one  of  the  two  parallel  con- 
taining planes  ;  indeed,  comparing  with  (1)  we  see  that  a  =  0,  6  =  ir,  c  =  0, 
so  that  this  case  is  such  an  extremely  simple  "  quadratic  "  that  it  actually 
reduces  to  a  linear  function,  since  a  =  0.     Since  this  results  favorably, 
the  prismoid  formula  applies.     It  is  easy  to  see  that 
B  =  0,    r=4ir,  M=2ir; 
B+T  +  A?I    .   _  0  +  4  TT  +  - 


hence 


T': 


4  =  8: 


which  agrees  with  the  result  of  Ex.  2,  §  70. 

Example  2,    The  prismoid  formula  applies  to  any  frustum  of  an  ellipsoid 
of  revolution  cut  off  by  planes  per- 
pendicular to  the  axis  of  revolution. 

Let  the  origin  be  situated  on  one 
of  the  truncating  planes  of  the  frus- 
tum, and  let  the  axis  of  x  be  the 
axis  of  revolution.  Then  the  equa- 
tion of  the  generating  ellipse  is  of  tl 
form  Ax'^  +  By-  +  Bx  +  F=0.  The 
area  As  of  a  section  parallel  to  the 
bases  is  tt*/-,  since  the  section  is  a 
circle  whose  radius  is  y.     Hence 

A 


iry- 


D         F\ 

B'^'-B'-Br 

which  is  a  quadratic  function  of  the  distance  x  from  one  of  the  truncating 
planes  of  the  frustum.     Therefore  the  prismoid  formula  holds. 

Beware  of  applying  the  prismoid  formula,  as  anything  but  an  approxi- 
mation formula,  without  knowiiig  that  the  area  of  a  section  is  a  quadratic 
function  of  s,  or  (Ex.  3,  p.  128)  a  cubic  function  of  s. 


128  INTEGRATION  [V,  §  71 

EXERCISES  XXVII.— GENERAL  EXERCISES 

[This  list   includes   a  number  of   exercises  which   are   intended   for 
reviews.] 

1.  Show  that  the  prismoid  formula  holds  for  each  of  the  following 
elementary  solids  ;  hence  calculate  the  volume  of  each  of  them  by  that 
formula:  (a)  sphere;  (6)  cone;  (c)  cylinder;  (d)  pyramid;  (e)  frus- 
tum of  a  sphere  ;   (/)  frustum  of  a  cone.     See  Tables,  II,  F. 

2.  Calculate  the  volume  of  the  solid  formed  by  revolving  the  area 
between  the  curve  y  =  x^  and  the  sc-axis  about  the  a'-axis,  between  x  =  0 
and  X  =  2.  Find  the  same  volume  (approximately)  by  the  prismoid 
formula,  and  show  that  the  error  is  about  4.2  %. 

3.  Calculate  the  volume  of  a  frustum  of  a  solid  bounded  by  planes 
h  =  0  and  h  =  H,\i  the  area  As  of  a  parallel  cross  section  is  a  cubic 
function  ali^  +  hi)?-  -\-  ch  -\-  d  of  the  distance  h  from  one  base,  first  by 
direct  integration,  then  by  the  prismoid  formula.  Hence  prove  the  state- 
ment of  the  footnote,  p.  125. 

4.  In  which  of  the  exercises  under  Exs.  4-9,  List  XXVI,  does  the 
prismoid  formula  give  a  precise  answer  ? 

5.  How  much  is  the  percentage  error  made  in  computing  the  volume 
in  Ex.  8  a,  List  XXVI,  from  x  =  l  to  x  =  3,  by  use  of  the  prismoid 
formula  ? 

6.  Show,  by  analogy  to  §  71,  that  the  area  under  any  curve  whose 
ordinate  y  is  any  quadratic  function  (or  any  cubic  function)  of  x,  between 
X-  a  and  x  =  &,  is 

^-^^IVA  +  ^yM  +  yBl, 

o 

where  yA,  Vb,  Vm  represent  the  values  of  y  at  .x  =  a,  x  =  &,  x  =  (a  -|-  6)/2, 
respectively. 

7.  Calculate,  first  by  direct  integration,  and  then  by  the  rule  of 
Ex.  6,  the  areas  under  each  of  the  following  curves  : 

(a)  y  =  X'  +  2x  -{■  Z  between  x  =  1  and  x  =  5. 
(&)  2/  =  x^  -l-^x  +  q  between  x  =  a  and  x  =  6. 
(c)    y  =  x^  -{■  bx  between  x  =  2  and  x  =  4. 

8.  Calculate  approximately  the  area  under  the  curve  y  =  x*  between 
x  =  l  and  X  =  3  by  the  rule  of  Ex,  6.     Show  that  the  error  is  about  .65%. 

9.  Show  that  the  area  under  the  curve  y  =  l/x^  between  x  =  1  and 
X  =  5  can  be  found  more  accurately  from  the  rule  of  Ex.  G  by  first  divid- 
ing the  area  into  two  parts  with  equal  bases. 


V,  §  71]  PRISMOID   FORMULA  129 

10.  Show  that  any  integral  whose  integrand  /(a-)  is  a  quadratic  (or  a 
cubic)  function  of  a:,  can  be  evaluated  by  a  process  analogous  to  the  pris- 
moid  rule  : 

11.  Evaluate  the  integral  I  {\/x-)dx  between  x  =  \  and  x  =  5  ap- 
proximately, first  by  the  rule  of  Ex.  10  ;  then  by  applying  the  same  rule 
twice  in  intervals  half  as  wide  ;  then  by  applying  the  rule  to  intervals  of 
unit  width. 

12.  Show  that  any  integral  {/(x^dx  can  be  computed  approximately 
by  using  Ex.  10  with  an  even  number  of  intervals  of  small  width  Ax  : 


£jy(^-)f?-> 


f{a)  +  4/(ff  +  A.r)  +  2/(a  +  2  Ax)  +  if  {a  +  3  Ax) 
Ax 


+  ■■'  +/(6) 


J   3' 


[This  rule  is  called  Simpson's  Rule ;  see  §  12.5.] 

13.  Calculate  the  following  integrals  approximately  by  the  process 
suggested  in  Exs.  11-12.  Notice  that  some  of  them  cannot  be  evaluated 
otherwise  at  present : 


(«)    Cx^dx.  (c)     CVxdx.  (e)     fvi  +  x^dx. 

(6)   £(l/x)dx.  (d)   j^'^VTT^fZx.  (/)  j^" 


sm  X  di 


14.  Show  that  the  area  of  any  surface  of  revolution,  formed  by  revolv- 
ing a  curve  y  =/(x)  about  the  x-axis,  is  the  limit  of  a  sum  of  terms  of  the 
form  2  Try  As,  where  s  denotes  the  length  of  arc,  as  in  §  61.  Hence  show, 
by  §  (31,  that  the  area  is  given  by  the  integral 


2.j-,*=2,j-,v^+(iy^. 


15.  In  a  manner  analogous  to  Ex.  14,  show  that  the  area  of  a  surface 
of  revolution  formed  by  revolving  a  curve  about  the  y-axis  is2ir  ixds. 

16.  Find  approximately  the  length  of  the  arc  of  the  curve  y  =  x^  from 
X  =  0  to  X  =  ^  ;  from  x  =  ^  to  x  =  1.     (See  Ex.  1,  p.  108.) 

17.  Find  approximately  the  area  of  the  convex  surface  of  that  portion 
of  the  paraboloid  formed  by  revolving  the  curve  y  =  Vx  about  the  x-axis 
which  is  cut  off  by  the  planes  x  =  0  and  x  =  ^  ;  by  x  =  J  and  x  =  1. 

K 


CHAPTER   VI 

TRANSCENDENTAL  FUNCTIONS 

PART   I.     LOGARITHMS  — EXPONENTIAL   FUNCTIONS 

72.  Necessity  of  Operations  on  Transcendental  Functions. 

The  necessity  for  the  introduction  of  transcendental  functions 
in  the  Calculus  depends  not  only  on  their  own  general  impor- 
tance, but  also  upon  the  fact  that  integrals  of  algebraic  functions 
may  he  transcendental. 

Thus,  in  §  57,  in  the  case  n  =  —  1  the  integral  J.r"  dx  could  not 
be  found,  although  the  integrand  1/x  is  comparatively  simple. 
We  shall  see  that  this  integral,  Ja;~'f/a;,  results  in  a  logarithm. 
(See  §  78,  p.  137,  Ex.  3.)  We  shall  see  also  in  §  81  that  nu- 
merous cases  arise  in  science  in  which  the  rate  of  variation  of 
a  function /(a;)  is  precisely  1/x. 

In  Ex.  1,  p.  108,  the  integral  J  VT+Ta?  dx  could  not  be 
evaluated  ;  throughout  Chapter  V,  integrals  involving  radicals 
were  avoided  except  in  special  cases,  because  such  integrals 
usually  result  in  transcendental  functions. 

73.  Properties  of  Logarithms.  The  logarithm  X  of  a  number 
N  to  any  base  B  is  defined  by  the  f a^t  that  the  two  equations 

(1)  N  =  B^,        logsN  =  L 

are  equivalent.  Thus  if  i  =  log^  N  and  I  =  log^  n,  the  identity 
B^  •  B^  =  5^+^  is  equivalent  to  the  rule 

(2)  log^  (^V .  n)  =  log^  N  +  log^  n, 

where  n  and  iVare  any  two  numbers.  Likewise  B^-~B^=B^~' 
gives 

(3)  log,(N^n)  =  \og,N-log,n; 

130 


VI,  §74]     LOGARITHMS  AXD   EXPONENTIALS  131 

and  (B'-y  =  B'"  becomes 

(4)  log^  ^Y"  =  n  log^  iV, 
where  n  may  have  any  value  whatever. 

Another  fundamental  rule  results  from  the  application  of  (4) 
to  the  equation 

(5)  x=B^,    i.e.   y=\ogj,x. 
For  if  h  is  any  other  base, 

(6)  log,  X  =  log, {By)  =  y  log,  B ;  [by  (4)] 
but  since  y  =  log^  x,  we  have 

(7)  log,x  =  logj,X'\og,B. 

In  particular  if  x  =  b,  since  log,&  =  1,  we  have 

(8)  1  =  log«  b  .  log,  B,  or  log,  i5  =  1  h-  log^  b. 

The  equations  (1),  (2),  (3),  (4),  (7),  (8)  are  the  fundamental 
rules  for  logarithms.    (See  Tables,  II,  A.) 

74.    Graphical  Representation.     A  fairly  accurate  graph  of 

the  equation 

(1)  y  =  \ogsX 

is  obtained  by  writing  the  equation  in  the  form 

(2)  X  =  B", 

and  plotting  a  few  points  given  by  taking  integral  (positive  and 
negative)  values  of  y.  Thus  y  =  0,l,  2,  •••,  —1,  —2,  •••  give 
a-  =  l,  B,  B-,  ■',  1/B,  l/S^,  •...  The  student  should  draw  a 
figure  from  such  values,  for  several  different  values  of  B,  taking 
5  =  2,  then  B  =  S,5,  10,  etc.  When  B  =  l,  the  equation  (2) 
degenerates  into  the  horizontal  straight  line  x  =  l,  while  (1) 
degenerates  completely  and  becomes  meaningless;  for  this 
reason,  tJie  number  1  is  never  vsed  as  a  base  of  logarithms. 

To  make  these  graphs  accurately,  more  points  are  necessary. 
The  easiest  method  is  to  calculate  the  desired  values  by  com- 
mon logarithms,  i.e.  logarithms  to  the  base  10.  Taking  the 
common  logarithms  of  both  sides  of  (2),  we  find 


132 


TRANSCENDENTAL   FUNCTIONS        [VI,  §  74 


logio  X  =  logjo  B'  =  y  •  logio  B, 

(3)     or  y  =  logio  x^  login  B. 

It  should  be  noticed  that  (3)  is  equivalent  to  (1)  and  therefore 
to  (2) ;  the  curves  for  B  =  1.5,  B  =  2,B  =  3,  B  =  4.5,  5  =  9  are 
shown  in  the  figure.  They  should  be  carefully  drawn  on  a 
much  larger  scale  by  the  student,  by  use  of  (3).     See  Tables. 


-1 

1 

^ 

— 

P 

^ 

P 

^ 

<^ 

- 

-4- 

y 

=  lo 

f^n 

X 

--' 

^ 

- 

^ 

J 

y 

^ 

— 

— 

' 

b 

_ 

^ 

^ 

/ 

^ 

■^ 

' 

,, 

s 

_ 

—i 

-J 

~ 

J 

^ 

, 

f 

4  5 

/ 

/ 

' 

__. 

. , 



^ 

/ 

y 

^ 

-^ 

•— 

2_ 

Lj 

. — ' 

— 

■p 

r 

^ 

•^ 

0 

f0 

1 

) 

3 

4 

: 

0 

/hi 

i 

I 

(\ 

Fig.  36. 


EXERCISES  XXVm.  -  LOGARITHMS  AND  EXPONENTIALS 

1.  Find  the  value  of  10^  when  x  =  2;  0  ;  1.5 ;  2.3  ;  -  1  ;  -  1.7  ;  0.43. 

2.  Plot  the  curve  y  =  10*  carefully,  using  several  fractional  values  of  x. 

3.  Plot  the  curve  y  =  logio  x  by  direct  comparison  with  the  figure  of 
Ex.  2.    Plot  it  again  by  use  of  a  table  of  logarithms. 

4.  Plot  the  graph  of  each  of  the  following  functions : 

(a)  logio  a;2.         (&)  logio  (1/x).         (c>  logio  Vx.        ((?)  logio  «2/8. 
Do  any  relations  exist  between  these  graphs  ? 

5.  Plot  the  graph  of  each  of  the  following  functions  and  explain  its 
relation  to  graphs  already  drawn  above  : 

(a)  logio  (1  + a;).         (t>)  logio  Vl  +  x.        (c)  logio  (xVl+x). 

6.  Plot  the  graphs  of  each  of  the  following  functions  and  show  the 
relations  between  them. 

(a)  logaa;.        (b)  logi  x.        (c)  logsxz.        (d)  2*. 


VI,  §  75]    DIFFERENTIATION   OF   LOGARITHMS  133 

7.  Show  how  to  calculate  most  readily  the  values  of  the  following 
expressions,  and  find  the  numerical  value  of  each  one : 

(a)  logiiT.  (c)    (5.4)6-2.  (e)    10>-5+ 10-1-5.         (g)  [ogslO. 

(6)  2*-53.  (d)  logs 8.  (/)   5  1og4  6.  (h)  lOiogio'. 

8.  Draw  each  of  the  following  curves  : 

(a)  y  =  10' +  10~'.  (c)  y  =  xlogiox.  (e)  y  =  logio  cos  ac. 

(6)  p^i-"  =  const.  (d)  y  =  2'  sin  x.  (/)  2/  =  10  ^'^^ 

75.    Slope  of  y  =  log^o  x  a,tjc  =  1.     The  slope  M  of  the  curve 

(1)  y  =  logioo; 

at  the  point   (1,  0)  can    be  approximated  very  closely.     Let 
(1,  0)  be  called  P,  and  let  (1  +  A.r,  0  +  Ay)  be  called  Q ;  then 

0  +  Ay=  logio  (1  +  Ax), 

and  the  slope  m^g  of  PQ  is 

(2)  m,Q  =  ^  =  ^-^^^^±M, 
'  ^      Ax  Ax 

If  Ax  is  given  in  succession  the  values  .1,  .01,  .001,  we  find 
'^^Jax=i  =  10  logio  (1-1)  =  0.4139; 
mpoj      ^  =  100  logio(1.01j  =  0.432} 

Wpg]    _^^^=  1000  logio  (1.001) 

=  0.43  [using  five-place  tables] 

=  0.434  [using  six-  or  seven-place  tables]. 

Still  smaller  values  of  Ax  would  give  the  same  result  by  the 
usual  interpolation  rules,  so  that  for  values  of  Ax  less  than 
.001  a  table  of  more  than  seven  places  would  be  needed ;  and 
even  then  the  result  would  be  changed  at  most  in  the  fourth 
place  of  decimals. 


134  TRANSCENDENTAL  FUNCTIONS        [VI,  §  75 

The  slope  M  oi  the  curve  (1)  at  (1,  0)  is  the  limit  of  these 
slopes  as  A.i;  approaches  zero ;  hence 

(3)  M=^^~\     =  lim  mpy  =  0.434  •••  (approximately).* 

76.  Differentiation  of  logjo^?.  It  is  now  easy  to  find  the 
derivative  of  logio  re.     Let  P,  (a;,  ?/),  be  any  point  for  which 

(1)  .    2/  =  logio  X,    o\  x  =  10^ ; 

and  let  Q,  {x  +  Aa;,  y  +  A?/),  be  any  other  point  on  the  curve ; 
then 

(2)  2/ +  A?/ =  logiu  (.T  +  Aa-),    or   a.- +  Ax  =  10"+-^^'. 

Subtracting  the  second  form  of  (1)  from  the  second  form  of  (2), 

Ax-  =  10"+^^  - 10^  =  10»'(10^^  -  1) 
and 

^  '  Ay  Ay  Ax      a;     lU^«'  - 1' 

since  a;  =  10«'. 

In  particular  at  x  =  1, 

Ay  _       A?/ 

which,  by  §  75,  approaches  the  limit  3/=  0.434  •••. 
In  general,!  therefore, 

(4)  flu  —  (Hogioa?  ^  j.jjj  ri  Ay      1  =  M  =  0.434  •»  ^ 

*  This  assumes  only  that  the  ordinary  interpolation  scheme  for  common 
logarithms  is  approximately  correct.  The  number  M  is  so  important  that  its 
value  has  been  calculated  to  a  large  number  of  decimal  places  ;  to  ten  places 
it  is  0.4342944819.  An  independent  method  of  calculating  it  is  given  in  §  134. 
Logically,  the  present  approximate  determination  of  M  could  be  omitted 
entirely  until  that  time,  and  M  could  be  carried  through  all  the  work  as  an 
unknown  constant.  Practically,  it  is  very  desirable  to  have  an  approximate 
value  of  M  at  once. 

t  The  difficulties  ordinarily  met  in  proving  this  formula  are  here  avoided 
by  placing  the  burden  of  any  difficulty  where  it  should  be,  —  upon  the  read* 


VI,  §  77]    DIFFERENTIATION   OF   LOGARITHMS  135 

77.  Differentiation  of  log^  jc.  Since  by  (3),  §  74,  the 
equation 

(1)  y  =  log  J,  X 
can  also  be  written  in  the  form 

(2)  y  =  logio  a;  -=-  logw  B, 
it  follows  that 

(3)  ^  = ^1^  ^ ^  ^  logjo  5  =  —  -=-  logio  5. 

dx  clx  dx  X 

Since  the  number  Af  which  occurs  in  all  these  formulas  is 
an  inconvenient  decimal,  it  is  useful  to  find  a  value  of  B,  for 
which 

(4)  logio i3  =  3/=  0.434  ••■; 

this  value  is  readily  found  from  a  logarithm  table,  and  is 
denoted  by  the  letter  e: 

(5)  e  =  10^  =  2.72  •••  (approximatelj). 
If  i^  =  e,  the  formula  (3)  becomes 

[VIII J  ^io&^  =  L. 

On  account  of  the  simplicity  of  this  formvla  the  base  e  will  be 
vsed  hoiceforth  in  this  book  for  all  logarithms  and  exponentials 
unless  the  contrary  is  exjilicitly  stated;*  it  is  called  the  natural 
base,  or  tlie  Napierian  base. 

If  B  has  any  value  whatever,  (3)  becomes 

[V..I]      'l^  =  l.-K^  =  Vj^«Ji  =  l.io,,e; 
dx  X     log  JO  B      X  logio  B      X 

ing  of  ail  ordinary  table  of  logarithms  :  for  the  essence  of  the  difficulty  lies 
in  the  lack  of  accuracy  of  the  usual  elementary  definition  of  logarithms.  No 
pretense  of  rigorous  logic  in  the  proof  of  (4)  is  justified  unless  a  proof  that 
the  common  logarithm  of  any  number  exists  is  given. 

*  The  value  of  e  to  ten  places  is  2.718281H285.  Another  method  of  com* 
puting  its  value  is  given  iu  §  134 ;  see  also  §  142. 


d\og,ox_ 

^M  _ 

logio  e  _ 

_  0.434  ... 

clx 

X 

X 

X 

136  TRANSCENDENTAL   FUNCTIONS         [VI,  §  77 

for  theoretical  purposes,  the  last  form  is  used;  for  practical 
computations,  the  next  to  the  last. 
If  5  =  10,  we  find 

[vni,] 

These  three  Rules,  of  which  [VIII]  is  the  general  form,  are 
added  to  the  list  of  seven  Rules  in  Chapter  III.  While  the 
common  base  10  is  exceedingly  convenient  for  computations, 
the  new  base  e  is  simpler  in  all  theoretical  discussions,  chiefly 
because  [VIII„]  is  simpler  than  [VIIIj]. 

Logarithms  to  the  base  e  are  called  natural,  or  Napierian,  or 
hyperbolic  logarithms.     See  Tables,  V,  C. 

78.  Illustrative  Examples.  We  may  now  combine  Rule 
[VIII]  with  [I]-[VII],  and  with  the  reverse  differentiation 
(integration)  formulas  of  Chapter  V. 

Example  1.     Given  y  =  logio(2  x^  +  3),  to  find  dy/dx. 
Method  1.     Derivative  notation.     Set  m  =  2  x^  +  3,  then 

di_dy     du  _  dloginu  _  d  (2  x'^  +  3)  _,M^^^_    4  Mx  ^ 
dx      du     dx  du  dx  u  2  x^  +  3 

Method  2.     Differential  notation. 

d2/ =  d  logxo(2  x-^  +  3)  =  ^-ilL_  d(2  x2  +  3)  =  ^i^  (Zx. 

Example  2.  Find  the  area  under  the  cm-ve  y  =  l/x  from  x  =  1  to 
X=:10: 

]x=10  /•^=10 1  ,  -1  x=10         ,  1 

=  (        ifZx  =  log.x|         =  log,  10  =  —!—  =  -. 
;r=l  Jx=l      X,  Ja-=1  ^  lOgj^e         M 

*  The  number  lege  10  =  1 -4- iV  =  2.302585  is  important  because  common 
logarithms  (base  10)  are  reduced  to  natural  logarithms  (base  e)  by  multiplying 
by  this  number,  since  log«iV=logio  -ZVx  loge  10.  Similarly,  natural  loga- 
rithms are  reduced  to  common  logarithms  by  multiplying  hy  M=  logio  e ;  since 
logio  iV  =  3/  -=-  loge  JV.  It  is  easy  to  remember  which  of  these  two  multipliers 
should  be  used  in  transferring  from  one  of  these  bases  to  the  other  by  remem- 
bering that  logarithms  of  numbers  above  1  are  surely  greater  when  e  is  used 
as  base  than  when  10  is  used. 


VI,  §  78]    DIFFERENTIATION   OF   LOGARITHMS  137 

Example  3.     If  the  rate  of  increase  dy/dx  of  a  quantity  y  with  respect 
to  X  is  1/x,  find  y  in  terms  of  x. 
Since  dy/dx  =  1/x, 

y  =  \  -'dx  =  logeX  +  c, 

where  c  is  a  constant,  — the  value  of  y  when  a;  =  1.  It  should  be  noted 
that  logarithms  to  the  base  e  occur  here  in  a  perfectly  natural  manner ; 
the  same  remark  applies  in  Example  2.     Note  that  loge  x  =  logio  x  -i-  M. 

This  case  arises  constantly  in  science.  Thus,  if  a  volume  v  of  gas  ex- 
pands by  an  amount  Ay,  and  if  the  work  done  in  the  expansion  is  A  W, 
the  ratio  A  ir/Au  is  approximately  the  pressure  of  the  gas;  and  dJI'/tZy 
=  p  exactly.  If  the  temperature  remains  constant  pv  =  a.  constant ;  hence 
dW/dv  =  k/v.     The  general  expression  for  W  is  therefore 


J   V 
expanding  from  one  volume 

'1'^='  =    (^'^  dv  =  k  loge  v\  "^  =    k  log.   ^-^  =  ^logjoH2. 


and  the  work  done  in  expanding  from  one  volume  vi  to  another  volume 
V2  is 

W 


EXERCISES  XXIX.  -  LOGARITHMS 

1.  Calculate  the  derivative  of  each  of  the  following  functions ;  when 
possible,  simplify  the  given  expression  first : 

(a)  logioa;2.  (6)  logio  Vx.  (c)  logio  (1+ Sx). 

{d)  logio  (1  +  x"-).  (e)  log,  (1  +  x)2.  (/)  log.  VrT2^. 

{g)  log.  (1/x).  {h)  logio  (x-2).  (0  a;log.x. 

c;)  log.  (i^^).  W  log.  (2  +  ^).       (0  log.  V^. 

\ra)'^2EA.  (u)  log, {log. x}.  (o)(log.O'^. 

2.  Evaluate  each  of  the  following  integrals  : 

■(^^r^-^'^-    wX'^*-     (oJ-;a-"-)(i  +  «-v» 


138  TRANSCENDENTAL   FUNCTIONS         [VI,  §  78 

3.  Calculate  the  area  between  the  hyperbola  ocy  =  I  and  the  a;-axis, 
from  a;  =  1  to  10,  10  to  100,  100  to  1000 ;  from  x=ltox  =  k. 

4.  Show  that  the  slope  of  the  curve  y  =  logio  x  is  a  constant  times  the 
slope  of  the  curve  y  =  log^  x.     Determine  this  constant  factor. 

5.  Find  the  flexion  of  the  curve  y  =  log, «,  and  show  that  there  are 
no  points  of  inflexion  on  the  curve. 

6.  Find  the  maxima  and  minima  of  the  curve  y  =  log^  (x^  —  2  x  +  3). 

7.  Find  the  maxima  and  minima  and  the  points  of  inflexion  (if  any 
exist),  on  each  of  the  following  curves: 

(a)  2/  =  2  a;2  -  log,  x.  (b)  y  =  x  +  log,  (1  +  a;-^). 

(c)  y  =  x^-  log,  x3.  (d)  2/  =  (2  X  +  log  x)'^. 

8.  Find  the  areas  under  each  of  the  following  curves  between  x  =  2 
and  X  =  5 : 

(a)  y  =  x  +  1/x.  (&)  y  =  (x^  +  l)/x3.  (f)  y  =  (xV2  _  x)/x^. 

9.  Find  the  volume  of  the  solid  of  revolution  formed  by  revolving  that 
portion  of  the  curve  xy-  =  1  between  x  =  1  and  x  =  3  about  the  x-axis. 
How  much  error  would  be  made  in  calculating  this  volume  by  the 
prismoid  formula  ? 

10.  If  a  body  moves  so  that  its  speed  v  =  t  +  l/t,  calculate  the  distance 
passed  over  between  the  times  t  =  2  and  t  =  4. 

11.  Find  the  work  done  in  compressing  10  cu.  ft.  of  a  gas  to  5  cu.  ft., 
ifjsy  =  .004. 

12.  Find  the  areas  under  the  hyperbola  xy  =  k^  between  x  =  1  and 
X  —  c,  c  and  c^,  c'^  and  c^,  c'  and  c*. 

79.   Differentiation  of  Exponentials.     Since  the  equations 
y  =  logg  X  and  a;  =  B'' 
are  equivalent,  Rule  [VIII]  gives 

^  =  ^=l-^^  =  -^_=5-log,B. 
ay       ay  ax     log^  e 

If  we  interchange  the  letters  x  and  y,  for  convenience  of 
memory,  we  obtain  the  standard  forms : 

y  =  B'^  (or  X  =  log^  y) 

dx       dx        log„e 


VI,  §  SO]  DIFFERENTIATION  OF  EXPONENTIALS        139 
of  which  the  two  special  cases  B  =  e  and  jB  =  10  are : 

This  formula  [IX]  can  be  combined  with  all  the  preceding 
rules,  as  in  §  78. 

80.   Illustrative  Examples. 

Example  1.     Given  y  =  e"-,  to  find  dy/dx. 
Method  1.     Set  x^  —  u  ;  then 

ax     du      dx      du        dx 
Method  2.  dy  =  de"""  =  e=^'  d(x^)  =  2  x  e*'  dx. 

Example  2.    Find  the  length  I  of  the  arc  of  the  catenary  y  =  (e*+e~')/2, 
between  the  points  where  a;  =  0  and  where  x  =  1. 
By  §  61,  p.  107,  we  have 


=  (•'=' Jl  +  (JIjuJ:^  dx  =  i  f  ^'  {e'  +  e-')  dx 

^Ife-l^^  (2.718 -0.368)72  =  1.175  (nearly). 
2\        e  / 

This  curve  is  very  important  because  it  is  the  form  taken  by  a  perfect 
inelastic  cord  hung  between  two  points.  Tlie  given  function  is  often 
called  the  hyperbolic  cosine  of  x,  and  is  denoted  by  cosh  cc,  so  that 
coshx  =(e*  +  e-^)/2. 

Example  3.  If  a  quantity  y  has  a  rate  of  change  dy/dx  with  respect 
to  X  proportional  to  y  itself,  to  find  y  in  terms  of  x.     Given 

dx 


140  TRANSCENDENTAL  FUNCTIONS        [VI,  §  80 

we  may  write 

dy     y' 
hence 


kx=  \  -  dy  =  loge 


+  0, 


by  §  78,  Ex.  3.     Transposing  c,  we  have 

loge  y  =  kx  —  c,    or    2/  =  e*^-"  =  g-^e*^  —  Ce**, 
where  C(=  e~'^)  is  again  an  arbitrary  constant. 

The  only  quantity  y  xohose  rate  of  change  is  proportional  to  itself  is 
Ce*^  where  O  and  k  are  arbitrary,  and  k  is  the  factor  of  proportionality . 
This  principle  is  of  the  greatest  importance  in  science  ;  a  detailed  dis- 
cussion of  concrete  cases  is  taken  up  in  §  81. 

EXERCISES  XXX.  —  EXPONENTIALS 

1.  Show  that  the  slope  of  the  curve  y  =  e^  is  equal  to  its  ordinate. 

2.  Show  that  the  area  under  the  curve  y  =  e^  between  the  y-axis  and 
any  value  oixis  y  —  \. 

3.  Find  the  derivative  of  each  of  the  following  functions  : 

(a)  e^-.  (d)(e-  +  l)'^.  ^„^   ^  -  e-'  (j)  e^-H*. 

(6)  e'\  2 

(c)  ire*.  2 

4.  The  expression  (e*—  e-^)/2,  used  in  Ex.  3  (/)  is  called  the  hyper- 
bolic sine  of  x ;  and  (e^  -\-  e"^)/2  is  called  the  hyperbolic  cosine  of  x  ; 
they  are  represented  by  the  symbols  sinh  x-  and  cosh  x  respectively. 
See  Tables,  II,  H,     Show  that 

d  sinh  ,T.  =  cosh  x  dx,     d  cosh  x  —  sinh  x  dx. 

5.  Show  that  1  -i-  sinh^x  =  cosh^a; ;  hence  find  the  length  of  the  arc  of 
the  curve  y  —  cosh  x  from  x  =  0  to  x  =  2. 

[The curve  2/  =  coshx,  ory  =  (e^  -f-  e-^)/2,  is  called  a  catenary  (§80).] 

6.  Find  the  area  under  the  catenary  from  x  =  0  to  x  =  3  ;  from 
x  =  — 1  to  x  =  -|-l;  from  x  =  0  to  x  =  a.     [See  Tables,  V,  C] 

T.  Find  the  area  under  the  curve  y  =  ainhx  from  x  =  0  to  x  =  3; 
from  X  =  0  to  x  =  a. 


""   e-  +  e-- 

W   I'+x^- 

{k)  10^'. 

(i)  x'^^K 

(0    X  •  102^+8 

VI,  §  81]  COMPOUND   INTEREST  LAW  141 

8.  Find  the  maxima  and  minima  and  the  points  of  inflexion  (if  any 
exist)  on  each  of  the  following  curves: 

(a)  y  =  sinh  x.         (6)  y  =  cosh  x.         (c)  y  =  tanh  x  =  sinh  x/cosh  x. 
(d)  2/ =  e-^^  (e)  y  =  e-^*.  (/)  y  =  sech  x  =  1 -^  cosh  x. 

9.  Show  that  the  pair  of  parameter  equations  x  =  cosh  t,  y  =  sinh  t 
represent  the  rectangular  hyperbola  x^  —  y-  =  l.  Hence  show  that  the 
differential  of  arc  for  this  hyperbola  is  ds=  (cosh  2  ty/- dt,  and  find  the 
speed  at  the  point  where  « =  0,  it  t  denotes  the  time. 

10.  Show  that  the  area  under  the  hyperbola  x^  —  y"^  =  I  from  x  =  1 
to  X  =  a  is  represented  by  the  integral 


£ 


sinh'-  tdt=  \       [(cosh  2  t  -  l)/2]  dt 


where  cosh  c  =  a.     Hence  show  that  this  area  is  (sinh  2  c)/4  —  c/2. 

11.  Show  that  the  area  of  a  triangle  whose  vertices  are  the  origin,  the 
point  (x,  0),  the  point  (x,  y)  on  the  hyperbola  x^  —  y^  —  1,  is  xy/2  = 
(sinh  2  0/4-  [Ex.  9.]  Hence  show  by  Ex.  10  that  the  portion  of  this 
triangle  outside  of  the  hyperbola  is  t/2. 

[Note.  The  parameter  equations  are  often  written  in  the  form 
X  —  cosh  2  A,  y  =  sinh  2  A,  where  A  is  the  last  area  mentioned.] 

12.  Calculate  the  following  integrals  : 

(a)  ]■/%'  + i)2dx. 

(Ii)    ( (e'  +  S)  e-' dx. 
(0  j {e'-^+^  +  1)  dx. 

81.   Compound  Interest  Law.     The  fact  proved  in  the  Ex.  3 
of  §  80  is  of  great  importance  in  science  : 
If  a  variable  quantity  y  has  a  rate  of  increase 

ivith  respect  to  an  independent  variable  x  proportional  to  y  itself, 

then 

(2)  y=Ce^, 

ichere  C  is  an  arbitrary  constant. 


(a)   £e^dx. 

((7)    rsiuh2xdx. 

(b)  |;.-^dx. 

(e)     CcoshSxdx. 

(c)   ^\^dx. 

(/)    Csinh'^xdx. 

142  TRANSCENDENTAL  FUNCTIONS        [VI,  §  81 

For  this  reason  the  equation  (2)  between  two  variables  x 
and  y  was  called  by  Lord  Kelvin  the  "Compound  Interest  Law," 
on  account  of  its  crude  analogy  to  compound  interest  on 
money.  For  the  larger  the  amount  y  (of  principal  and  in- 
terest) grows  the  faster  the  interest  accumulates. 

"  Compound  interest "  is,  however,  only  a  convenient  name, 
since  interest  is  really  compounded  at  stated  intervals  (e.g.  each 
year)  and  not  continuously.  A  more  suggestive  name  might 
be  the  snowball  law,  since  a  snowball  grows  more  rapidly  the 
larger  it  becomes,  and  its  rate  of  growth  is  roughly  propor- 
tional to  its  size. 

In  science  instances  of  a  rate  of  growth  which  grows  as  the 
total  grows  are  frequent.* 

Example  1.  Work  in  Expanding  Gas.  The  example  used  to  illns- 
trate  Ex.  3,  §  78,  can  be  put  in  this  form.  Since,  in  the  work  W  done  in 
the  expansion  at  constant  temperature  of  a  gas  of  volume  -y,  we  found 
dW/dv  =  k/v,  it  follows  that  dv/dW=v/k;  hence  v  =  Ae^/'',  which 
agrees  with  the  result  of  §  78. 

Example  2.  Cooling  in  a  Moving  Fluid.  If  a  heated  object  is  cooled 
in  running  water  or  moving  air,  and  if  6  is  the  varying  difference  in 
temperature  between  the  heated  object  and  the  fluid,  the  rate  of  change 
of  d  (per  second)  is  assumed  to  be  proportional  to  6  ;  dd/dt  =  —  kd,  where 
t  is  the  time  and  where  the  negative  sign  indicates  that  d  is  decreasing. 
It  follows  that  0=0-  e"*'.     [Newton's  Law  of  Cooling.] 

Such  an  equation  may  also  be  thrown  in  the  form  of  §  78  ;  in  this 
example,  dt/dS  =—  l/{kd),  whence  t  =  —  ('i./k)  ■  log^  0  +  c,  and  the  time 
taken  to  cool  from  one  temperature  0i  to  another  temperature  02  is 

Je=e,     Je,      k0        k    °"   je^         k    ^'^i 

where  0  is  the  temperature  of  the  body  above  the  temperature  of  the 
surrounding  fluid. 


*  The  common  expressions  "grows  like  a  snowball,"  "gathers  momentum 
as  it  goes,"  "wealth  breeds  wealth,"  "it  grows  by  its  very  growth,"  "the 
rich  grow  richer,  the  poor  poorer"  illustrate  the  frequent  occurrence  of  such 


VI,  §81]  COMPOUND   INTEREST    LAW  143 

The  law  for  the  dying  out  of  an  electric  current  in  a  conductor  when 
the  power  is  cut  off  is  very  similar  to  the  law  for  cooling  in  this  example. 
See  Ex.  17,  p.  146. 

Example  3.  Bacterial  Growth.  If  bacteria  grow  freely  in  the  pres- 
ence of  unlimited  food,  the  increase  per  second  in  the  number  in  a  cubic 
inch  of  culture  is  proportional  to  the  number  present.     Hence 

'—  =  kX,  N-  =  Ce",  t  =  -  log,  .V  +  c, 
dt  A     °  ' 

where  iVis  the  number  of  thousand  per  cubic  inch,  t  is  the  time,  and  k 
is  the  rate  of  increase  shown  by  a  colony  of  one  thousand  per  cubic  inch. 
The  time  consumed  in  increase  from  one  number  iVi  to  another  number 

tT=  ri^=iiog..v7^=iiog.^. 

\     J.Yj  k  iV      k    *      J.\     k        Xi 

If  iV2  =  10j\"i,  the  time  consumed  is  (1/A-)  loge  10  =  l/(i.V).  This 
fact  is  used  to  determine  k,  since  the  time  consumed  in  increasing  iV  ten- 
fold can  be  measured  (approximately).  If  this  time  is  T,  then  T  =  \/(JcM), 
whence  k  =  1/{TM),  where  Tis  known  and  M  =  0.43  (nearly). 

Numerous  instances  similar  to  this  occur  in  vegetable  growth  and  in 
organic  chemistry.  For  this  reason  the  equation  (2)  on  p.  141  is  often 
called  the  "law  of  organic  growth."     (See  Exs.  18,  19,  p.  146.) 

Example  4.  Atmospheric  Pressure.  The  air  pressure  near  the  surface 
of  the  earth  is  due  to  the  weight  of  the  air  above.  The  pressure  at  the 
bottom  of  1  cu.  ft.  of  air  exceeds  that  at  the  top  by  the  weight  of  that 
cubic  foot  of  air.  If  we  assume  the  temperature  constant,  the  volume  of 
a  given  amount  is  inversely  proportional  to  the  pressure,  hence  the  amount 
of  air  in  1  cu.  ft.  is  directly  proportional  to  the  pressure,  and  therefore 
the  weight  of  1  cu.  ft.  is  proportional  to  the  pressure.  It  follows  that  the 
rate  of  decrease  of  the  pressure  as  we  leave  the  earth's  surface  is  propor- 
tional to  the  pressure  itself : 

^P=-kp,  p=  Ce-*\    h=-'^  \og,p  +  c, 
dh  k 

where  h  is  the  height  above  the  earth ;  and,  as  in  Exs.  2  and  3,  the  dif- 
ference in  the  height  which  would  change  the  pressure  from  pi  to  p^  is 

Since  /i]p^  and  p2  and  pi  can  be  found  by  experiment,  k  is  determined 
by  the  last  equation. 


144  TRANSCENDENTAL  FUNCTIONS        [VI,  §  82 

82.  Percentage  Rate  of  Increase.  The  principle  stated  in 
§  81  may  be  restated  as  follows :  In  the  case  of  bacterial 
growth,  for  example,  while  the  total  rate  of  increase  is  clearly- 
proportional  to  the  total  number  in  thousands  to  the  cubic 
inch  of  bacteria,  the  percentage  rate  of  increase  is  clearly 
constant. 

In  any  case  the  percentage  rate  of  increase,  r^,,  is  obtained 
by  dividing  100  times  the  total  rate  of  increase  by  the  total 
amount  of  the  quantity,  100  •  (dy/clx)  -j-y;  and  since  the  equa- 
tion dy/dx=  ley  gives  (dy/dx)-^y  =  1c,  it  is  clear  that  the  per- 
centage  rate  of  increase  in  any  of  these  ptrohlems  is  a  constant. 
The  quotient  (dy/dx)  -f-  y,  that  is,  1/100  of  the  percentage  rate 
of  increase,  will  be  called  the  relative  rate  of  increase,  and 
will  be  denoted  by  7\. 

In  some  of  the  exercises  which  follow,  the  statements  are 
phrased  in  terms  of  percentage  rate  of  increase,  r^,  or  the  rela- 
tive rate  of  increase,  r^  =  ?"p  -r- 100. 

EXERCISES   XXXI.  — COMPOUND  INTEREST  LAW 

1.  If  2/  =  5  e^^,  find  chj/dx,  and  show  that  {dy/dx)  ^  y  =  2. 

2.  Find  dy/dx  and  (dy/dx)  -=-  y  for  each  of  the  following  functions  : 
(a)  7e3z.  (fZ)  e'\  (g)  (ax  +  6)e*^. 
(6)  4e-2-5«.                        (e)  e*'+\  (A)  {x^+px  +  q)e^. 
(c)  xe'.                             if)   (a;2  +  2)e».  (i)  (Sx  +  2)e-^\ 

3.  If  a  body  cools  in  moving  air,  according  to  Newton's  law,  dd/dt 
=  _  Jc0^  where  t  is  the  time  (in  seconds)  and  0  is  the  difference  in  tempera- 
ture between  the  body  and  the  air,  find  k  if  6  falls  from  40°  C.  to  30°  C. 
in  200  seconds. 

4.  How  soon  will  the  difference  in  temperature  e  in  Ex.  3,  fall  to 
10^  C? 

5.  If  a  body  is  cooled  in  air,  according  to  Newton's  law,  find  k  if  the 
0  changes  from  20°  C.  to  10°  C.  in  five  minutes.  How  soon  will  d  reach 
5°C.?^ 

6.  If  a  body  cools  so  that  the  percentage  rate  of  cooling  is  2  %  (in  de- 
grees C.  and  minutes),  how  long  will  it  talce  to  cool  from  a  difference  20° 
to  a  difference  10°  (with  respect  to  the  surrounding  air)? 


VI.  §  82]  CO:^IPOUND   INTEREST   LAW  145 

7.  In  measuring  atmospheric  pressure,  it  is  usual  to  express  the  pres- 
sure in  millimeters  (or  in  inches)  of  mercury  in  a  barometer.  Find  C  in 
the  formula  of  Ex.  4,  §  81,  ii  p  =  762  mm.  when  h  =0  (sea  level).  Find 
Cifp  =  SO  in,  when  h=:0. 

8.  Using  the  value  of  C  found  in  Ex.  7,  find  k  in  the  formula  for  at- 
mospheric pressure  if  p  =  24  in.  when  h  —  5830  ft.  ;  if  p  =  600  mm.  when 
h  =  1909  m.  Hence  find  the  barometric  reading  at  a  height  of  3000  ft. ; 
1000  m.     Find  the  height  if  the  barometer  reads  28  in.  ;  650  mm. 

[Note.  Pressure  in  pounds  per  square  inch  —  0.4908  x  barometer 
reading  in  inches.] 

9.  If  a  rotating  wheel  is  stopped  by  water  friction,  the  rate  of  decrease 
of  angular  speed,  dw/dt,  is  proportional  to  the  speed.  Find  w  in  terms  of 
the  time,  and  find  the  factor  of  proportionality  if  the  speed  of  the  wheel 
diminishes  50  %  in  one  minute. 

10.  If  a  wheel  stopped  by  water  friction  has  its  speed  reduced  at  a  con- 
stant rate  of  2%  (in  revolutions  per  second  and  seconds),  how  long  will 
it  take  to  lose  50  %  of  the  speed  ? 

11.  The  length  I  of  a  rod  when  hfeated  expands  at  a  constant  rate  per 
cent  ( =  100  k).  Show  that  dl/dO  =  kl,  where  6  is  the  temperature  ;  if  the 
percentage  rate  of  increase  is  .001  %  (in  feet  and  degrees  C),  how  much 
longer  will  it  be  when  heated  200°  C.  ?  At  what  temperature  will  the  rod 
be  1 7o  longer  than  it  was  originally  ? 

[Note.     This  value  of  k  is  about  correct  for  cast  iron.] 

12.  The  coefficient  of  expansion  of  a  metal  rod  is  the  increase  in 
length  per  degree  rise  in  temperature  of  a  rod  of  unit  length.  Show  that 
the  coeflScient  of  expansion  of  any  rod  is  the  relative  rate  of  increase  in 
length  with  respect  to  the  temperature.     (See  Ex.  12,  p.  27.) 

13.  A  chimney  is  designed  so  that  the  pressure  per  square  inch  on  each 
horizontal  cross  section  is  a  constant  k.  If  the  outer  surface  of  a  section 
at  a  height  h  is  a  circle  of  radius  B,  and  if  all  the  cross  sections  are  simi- 
lar, including  the  flue  holes,  show  that  the  total  pressure  on  a  cross  sec- 
tion is  proportional  to  kE-,  and  that  k(B  +  AR)'^  =  kB'^  -  pR-Ah,  where 
p  is  the  weight  per  cubic  inch  of  the  material.  Hence  show  that  dB/dh  = 
—  pB/{2  k)  and  that  B  =  i?oe"P*'''-*\  where  B^  is  the  radius  of  tiie  bottom 
section  {h  =  0). 

14.  Assuming  that  the  form  of  a  chimney  is  given  by  the  equation 
i?  =  i?oe"P*'^'*'  [Kx.  13],  show  that  the  total  weight  (neglecting  the  flue 
holes)  is  kTBQ-{l  —  e-p"''') ,  where  H  is  the  total  height.     Hence  .show  that 

L 


146  TRANSCENDENTAL   FUNCTIONS  VI,  §82 

the  pressure  per  square  inch  on  the  bottom  section  is  k(l  —  e-p^"'),  and 
that  it  approaches  the  theoretical  limit  A;  as  if  increases. 

15.  Show  that  the  results  of  Ex.  14  are  the  same  when  the  flue  holes 
are  taken  into  account,  with  the  assumptions  made  in  Ex.  13. 

[Note,  The  pressure  per  square  inch  depends  solely  on  the  height,  for 
the  same  material.  The  height  is  limited  by  the  crushing  strength  of  the 
material.] 

16.  When  a  belt  passes  around  a  pulley,  if  T  is  the  tension  (in  pounds) 
at  a  distance  s  (in  feet)  from  the  point  where  the  belt  leaves  the  pulley, 
)•  the  radius  of  the  pulley,  and  /j.  the  coefficient  of  friction,  then  dT/ds  = 
fiT/r.  Express  T  in  terms  of  s.  It  T=  30  lb.  when  s  =  0,  what  is  T 
when  s  =  5  ft.,  if  r  =  7  ft.,  and  /i  =  0.3  ? 

17.  "When  an  electric  circuit  is  cut  off,  the  rate  of  decrease  of  the  cur- 
rent is  proportional  to  the  current  C.  Show  that  C  =  Coe-**,  where  Co 
is  the  value  of  C  when  t  =  0. 

[Note.  The  assumption  made  is  that  the  electric  pressure,  or  electro- 
motive force,  suddenly  becomes  zero,  the  circuit  remaining  unbroken. 
This  is  approximately  realized  in  one-portion  of  a  circuit  which  is  short- 
circuited.  The  effect  is  due  to  self-induction  :  k  =  B/L,  where  B  is  the 
resistance  and  L  the  self-induction  of  the  circuit.] 

18.  Radium  automatically  decomposes  at  a  constant  (relative)  rate. 
Show  that  the  quantity  remaining  after  a  time  t  is  q  =  qoe~^,  where  go  is 
the  original  quantity.  Find  k  from  the  fact  that  half  the  original  quantity 
disappears  in  1800  yrs.    How  much  disappears  in  100  yrs.  ?  in  one  year  ? 

19.  Many  other  chemical  reactions  —  for  example,  the  formation  of  in- 
vert sugar  from  sugar  —  proceed  approximately  in  a  manner  similar  to 
that  described  in  Ex.  18.  Show  that  the  quantity  which  remains  is 
q  =  qoe-'^  and  that  the  amount  transformed  is  A=:  qo  —  q  =  qo{^  —  c— **). 
Show  that  the  quantities  which  remain  after  a  series  of  equal  intervals  of 
time  are  in  geometric  proportion. 

20.  The  amount  of  light  which  passes  through  a  given  thickness  of 
glass,  or  other  absorbing  material,  is  found  from  the  fact  that  a  fixed  per 
cent  of  the  total  is  absorbed  by  any  absorbing  material.  Express  the 
amount  which  will  pass  through  a  given  thickness  of  glass. 

83.  Logarithmic  Differentiation.    Relative  Increase.    In  §  82 

we  defined  the  relative  rate  of  increase  ?v  of  a  quantity  y  with 
respect  to  x  as  the  total  rate  of  increase  (dy/dx)  divided  by  y. 


VI,  §84]  RELATIVE  RATES  147 

If  y  is  given  as  a  function  of  x, 

(1)  y=A^), 

the  relative  rate  of  increase  ?v  =  idy/dx)  -=-  y  can  be  obtained 
by  taking  the  logarithms  of  both  sides  of  (1),* 

(2)  \o^,y  =  \og,f{x), 

and  then  differentiating  both  sides  with  respect  to  x-. 

(3)  r   =l-.*Iy=  ^^  ^^^^  y  =  dlogeAx) 

^      ij    dx^  dx  dx 

This  process  is  often  called  logarithmic  differentiation :  the 
logarithmic  derivative  of  a  function  is  its  relative  rate  of  increase, 
Tr,  or  1/100  of  its  percentage  rate  of  increase. 

Example  1.  Given  y  =  Ce>^,  to  find  Vr  =  {dy/dx)  -4-  y.  Taking  log- 
aritlims  on  both  sides : 

loge  y  =  loge  C  +  kx; 
differentiating  both  sides  with  respect  to  x  ; 

dx  dx 

The  resnlt  of  Ex.  3,  p.  139,  may  be  restated  as  follows:  the 
only  function  of  x  whose  relative  rate  of  change  (logarithmic 
derivative)  is  constant  is  Ce'". 

Example  2.    Given  j/  =  x^  +  3  a;  +  2,  to  find  r^. 

Method  1.  ^  =  2  X  +  3,  hence  r^  =  -^  -^  2a;  +  3 

dx  dx      ^      xi  +  ^x  +  2 

Method2.    .^  =  ^^^2/=^Mi/=.^nogixi+3x  +  2)^_lx+3_, 
'      dx  dx  dx  x2  +  3  X  +  2 

84.  Logarithmic  Methods.  The  process  of  logarithmic  dif- 
ferentiation is  often  used  apart  from  its  meaning  as  a  relative 
rate,  simply  as  a  device  for  obtaining  the  usual  derivative. 
This  is  particularly  useful  in  the  case  of  variables  raised  to 
variable  powers,  and  it  is  at  least  convenient  in  such  other 
examples  as  those  which  follow. 

*  Since  log  ^is  defined  only  for  positive  values  of  N,  all  that  follows  holds 
only  for  positive  values  of  the  quantities  whose  logarithms  are  used. 


148  TRANSCENDENTAL   FUNCTIONS        [VI,  §  84 

Example  1.     Given  y  =  Vx,  to  find  dy/dx. 
Method  1.    Ordinary  Differentiation. 

dy^  _  dVx  _  dx^^'  _  1      1/2  _     1     ^ 
dx~    dx   ~    dx    ~  2  ~  2  x^'^ ' 

Method  2.  Logaritlimic  Method. 
Since  for  positive  values  of  Vx,  « 

log  y  =  log  x'^^  =  ^  log  X, 
we  have 

Example  2.       Given  ?/  =  (2  a;2  +  3)  10*^-^ . 
Method  1.    Ordinary  Differentiation. 

^  =  (2  a;2  +  3)  f  (lO*^"')  +  lO^-i  A  (2  x2  +  3) 
dx  dx  dx  '^ 

=  (2x2 +  3). 4.  ll0'"-Vl0*""'-4x 

=  4  .  10i^-ir(2x2  +  3)/Jtf  +  x],  where  itf  =  logioe  =  0.434. 

ilfeifeoc?  2.    Logarithmic  Method. 

Since     log  y  =  log  (2  x2  +  3)  +  (4x-l)logl0, 

we  have  r,.  =  ^^y=     ^^      +  4  •  log  10, 

(Zx  2x-'  +  3  °      ' 

or  ^  =  y  r_i^_  +  4  log  lOl  =  4  .  10*^-^  [x  +  (2  x2  +  3)log  10], 

dx        L2  x2  +  3  J 

which  agrees  with  the  preceding  result,  since  loge  10  =  1/logio  e  =  \/M. 

Example  3.   Given  y  =  (S  x~+  1)2*+*,  to  find  dy/dx.    Since  no  rule  has 
been  given  for  a  variable  to  a  variable  power,  ordinary  differentiation 
cannot  be  used  advantageously.    Taking  logarithms,  however,  we  find 
log?/=(2x  +  4)log(3x2  +  lX 

whence  r;- =  ^  ^  y  =  2Iog  (3x2  +  1)  +      ^^     (2x  +  4), 

dx  3  x-  +  1 

or  ^  =(3^2  + i)2x+4J21og(3x2 +  l)  +  ;^4^  (2^  +  4)!  • 

dx  (  3  x^  +  1  ] 

The  use  of  the  logarithmic  method  is  the  only  expeditious  way  to  find 

the  derivative  in  this  example. 


VI,  §  84]  RELATIVE  RATES  140 

EXERCISES   XXXn.  —  LOGARITHMIC   DIFFERENTIATION 

1.  Find  the  logarithmic  derivatives  (relative  rates  of  increase)  of  each 
of  the  following  functions,  by  each  of  the  two  methods  of  §§  82-83: 

(a)  e-2«.  (e)  0.1  gio'-s.  (i)   (r^  +  i)  e-'-=. 

(6)  4e«  (/)  102-+3.  (j)   (2  -  3  «2)  c2<2-i. 

(c)  e»+2.  (gf)  e-«^+*^'.  (A,-)   (1  _  <■-'  +  <4)  lo^'+s^. 

(d)  e-»*.  (A)    2f%-5'.  (Z)     ger. 

2.  Find  the  derivative  of  each  of  the  following  functions  by  the 
logarithmic  method : 

(a)   (l  +  a:)i+x.  (c)  x^.  (e)   (1  +  x)(l  +  2  a:)(l  +  3  x). 

3.  If  y  =  WW,  show  that  dy  -^  y  =  du  ^  ti  +  dv  -i-  v.  In  general  show 
that  the  relative  rate  of  increase  of  a  product  is  the  sum  of  the  relative 
rates  of  increase  of  the  factors. 

4.  If  a  rectangular  sheet  of  metal  is  heated,  show  that  the  relative 
rate  of  increase  in  its  area  is  twice  the  coefficient  of  expansion  of  the 
material  [see  Ex.  12,  List  XXXI]. 

5.  Extend  the  rule  of  Ex.  3  to  the  case  of  any  number  of  factors. 
Apply  this  to  the  expansion  of  a  heated  block  of  metal. 

6.  Show  directly,  and  also  by  use  of  Ex.  5,  that  the  relative  rate  of 
increase  of  x"  with  respect  to  x,  where  n  is  an  integer,  is  n/x. 

7.  Compare  the  functions  e^^  and  e2i+3 .  compare  their  relative  rates 
of  increase  ;  compare  their  derivatives ;  compare  their  second  derivatives. 

8.  Compare  the  following  pairs  of  functions,  their  logarithmic  deriv- 
atives, their  ordinary  derivatives,  and  their  second  derivatives: 

(a)  e^  and  10^.  {d)  e-"^  and  6+"^. 

(6)  e"  and  6"+*.  (e)  e-*^  and  sech  x. 

(c)  e«  and  10"^.  (/)  e-^'  and  1  ^  (a  +  hx?). 

9.  Can  k  be  found  so  that  ke"  and  lO*"^  coincide  ?  Prove  this  by  com- 
paring their  logarithmic  derivatives,  and  find  h  in  terms  of  a. 

10.  If  the  logarithmic  derivative  {dy/dx)  ^  y  is  equal  to  3  +  4  x,  show 
that  log  2/  =  3  a;  +  2  X'^  +  const.,  or  y  =  ke^^+^'^. 

11.  If  {dy/dx)  ^y=f{x)  show  that  y  =  ke^^^'^'"'. 

12.  Find  y  if  the  logarithmic  derivative  has  any  one  of  the  following 
values : 

(a)  1  -  X.  (c)  n/x.  (e)  e'. 

(b)  ax  +  6x2.  (d)  a  +  n/x.  (/)  e*  +  n/x. 


150 


TRANSCENDENTAL  FUNCTIONS        [VI,  §  85 


Fig.  37. 


(1) 


PART   11.     TRIGONOIMETRIC   FUNCTIONS 

85.   Introduction  of  Trigonometric  Functions.     The  way  in 

which  trigonometric  functions  enter  in  the  Calculus  is  illus- 
trated by  the  following  simple  case  of 
uniform  rotation : 

A  point  Jfmoves  with  a  constant  speed 
of  1  ft.  per  second  on  a  unit  circle.  Let 
0  be  the  center  of  the  circle,  and  let  x 
and  y  be  the  horizontal  and  vertical  dis- 
tances, respectively,  of  the  moving  point 
M  from  0.  The  equation  of  the  circle 
ic^  +  2/2  =  1 

may  also  be  written  in  parameter  form 

(2)  x  =  cos^,     ?/  =  sin^, 
where  6  =  Z.  XOM,  as  is  evident  from  the  figure. 

If  0  is  measured  in  circular  measure,  6  =  s,  where  s  is  the 
arc  AM,  since  the  radius  is  1.  Moreover,  since  M  is  moving 
with  a  constant  speed  of  1  ft.  per  second,  s  =  t,  where  t  is  the 
time  measured  in  seconds  since  Jf  was  at  A,  and  s  is  measured 
in  feet.     The  equation  (2)  may  be  written  in  the  form : 

(3)  X  =  cos  t,     y  =  sin  t,  (where  $  =  s  =  t). 

The  horizontal  speed  of  M,  v^,  and  its  vertical  speed,  Vy,  are, 
respectively : 

(4) 


dx     dcost 


dy 
dt' 


d  sin  t 


dt  dt  "      dt  dt 

to  find  these  we  need  precisely  to  know  the  derivatives  of  cos  t 
and  of  sin  t  with  respect  to  t. 

86.  Bifferentiation  of  Sines  and  Cosines.  These  derivatives 
may  be  found  directly  from  the  example  of  §  85. 

To  do  so,  we  need  to  find  two  eqtiations  for  the  two  xmlinown 
quantities  dx/dt  and  dy/dt;  one  of  these  is  given  by  differen- 
tiating (1),  §  85,  with  respect  to  t : 


VI,  §  86]  TRIGONOMETRIC  FUNCTIONS  151 

the  other  is  found  from  the  fact  that  the  sum  of  the  squares 
of  v^  and  v^  is  equal  to  the  square  of  the  total  sj^eed  (§  62) : 


(fJHIJ 


=  1, 


since  the  speed  ds/dt  =  1  in  §  85.  Either  unknown  can  now 
be  found  by  solving  (1)  and  (2)  simultaneously : 

-ox  dx  •    .     dy       ,  ,          . 

(3)  —  =  -y  =  -s\nt,    -^  =  +  .T  =  +  cos^ 

dt  dt 

since  x^  +  y'  =  1.  In  extracting  square  roots  in  this  solution 
the  negative  sign  is  attached  to  the  value  of  dx/dt  because  x  is 
decreasing  when  y  is  positive.  The  signs  in  (3)  are  easily  seen 
to  be  correct  for  both  positive  and  negative  values  of  x  and  y. 
Comparing  (3)  with  equation  (4)  of  §  85,  we  find: 

[XI]      ^^  =  -smt,  [X]      ^^«  =  +  co8«, 

or,  in  differential  notation : 

[XI]'     dcos<  =  — sinf<«,  [X]'     rf  sin «  =  +  cos  « rt«. 

These  two  formulas  are  the  basis  of  all  work  on  trigonometric 
functions.  Circidar  measure  of  angles  teas  used  in  obtaining 
them,  and  this  system  of  measurement  will  be  used  oi  all  that 
follou's* 

A  direct  proof  of  these  two  important  formulas  is  easily 
made.     For,  let  y  =  sinx;  then  y  +  Ay  =  sin(.r  +  Ax), 

3^  +  -^  jsin— . 

Hence     ^  =  cos(x  +  ^)  •  5^1^^, 
Ax  V  2  /  Ax/2 

*  Circular  measure  of  angles  is  used  in  the  Calculus  for  the  same  reason 
that  Napierian  Logarithms  are  used  for  logarithmic  and  exponential  functions: 
in  each  case  the  standard  formulas  for  differentiation  are  simplest  in  the 
system  adopted. 


152 


TRANSCENDENTAL  FUNCTIONS        [VI, 


whence 


dy 
dx 


=  lim 


A?/ 


Ai=y)  Ax 

lim(sin  «)/«  =  !. 


d  sin  X 
:  cos  X,  or    — : =  cos  X, 


dx 


The  proof  of  [XI]  is  exactly  analogous.  See  also  Ex.  7, 
p.  154. 

87.  Illustrative  Examples.  The  formulas  [X]  and  [XI] 
may  be  combined  with  other  standard  formulas.  Some  of  the 
results  are  themselves  worthy  of  mention  as  new  standard 
formulas;  these  are  numbered  below  in  Roman  numerals  and 
printed  in  black-faced  type. 

Example  1.     Given  y  —  sm2  6,  find  dy. 

dy  =  d  (sin  26')  =  cos  2ed(2  0)  =  2  cos  2ed9. 
Example  2.     Given  y  —  tan  6,  find  dy/dd. 


dd 


[XII] 


J  sin  ^      „  „  o  f?  sin  0 

d cos  6 

cos  e  dd 


.    dd 
diand 


dO 


1 

C0S2^ 


C0S2^ 

sec2^. 


,  d  COS  6 

dd      _      1 

~C0S2d' 


Example  3.     Given  y  =  ctn  e,  to  find  dy/dd. 
,cos^ 


[XIII] 

Similarly, 

[XIV] 

[XV] 

Example  4. 

dy  _  de"    g, 
dx      dx 


d  ctn  0 

~~dd~' 

ds&cO 

'  dd 

descO 


sin^ 
dO 


dd 

,    1 

sin^ 


1 
sin^ 


sm 
cos- 


=  sec  9  tan  0. 


—  cos< 


=  —  CSC  ^  ctn  6. 


dd  dO  sin-  tf 

Given  y  =  e"  sin  x,  to  find  dy/dx. 

X  +  ^  ^'"  -^  .  e*  =  e*  sin  x  +  e*  cos  x  =  e*(sin  x  4-  cos  x). 
dx 


VI,  §87]  TRIGONOMETRIC   FUNCTIONS  153 

Example  6.     Given  y  =  cos^  (2  ("-  +  1),  to  find  dy/dt. 
Let  u  =  2  t-  +  1,  aud  v  =  cos  m,  then 

dt         dl  dt  dt  dt  dt 

z=  —  3  1)2  •  sin  M  ■  4  «  =  —  12  ( •  sin  (2  «^  +  1)  •  cos-  (2  t-  +  \). 

Example  6.  To  find  the  area  under  the  curve  y  =  sin  x  from  the  point 
where  x  =  0  to  the  point  where  x  =  7r/2. 

]x=ir/2  rx=JT/2  -\x=Tr/2 

=  \  sin  x  dx  =  —  cos  x  \  =—  cos  7r/2  +  cos  0  =  1, 

x=0  Jx=0  Jx=0 

since  d  (  —  cos  a;)  =  sin  x  dx.  Comparatively  few  of  the  trigonometric  in- 
tegrals can  be  found  by  simple  inspection  ;  a  detailed  treatment  of  them 
is  given  in  Chapter  VII. 

EXERCISES  XXXin.  — TRIGONOMETRIC  FUNCTIONS 

1.  Find  the  derivative  of  each  of  the  following  functions  : 

(a)  sinSx.  (e)  sinx^.  (i)   xsinx. 

(b)  cos  ((9/2).  (/)  tan(2<  +  3).  (j)   e'tan«. 

(c)  tan(— e).  (g)  cos  (—it).  {k)  log  cos  x. 

(d)  cos^x.                          (h)  sec(x/3).  (0    sin  e^. 
(«i)  sin  X  +  3  cos  2  X.                             (p)  e-'cos- (1  +  3«), 
00  e-' sin  (2  «  +  tt/IO)  .                        (ry)  e-^+^sm(3t-ir/i). 
(o)  (1 +  x2)sin(2x  +  3).                   {>•)  e-'/iO[cos«  +  4sin3<]. 

2.  Find  the  area  under  the  curve  y  =  sin  x  from  x  =  0  to  x  =  tt  ;  test 
the  correctness  of  your  result  by  rough  comparison  with  the  circumscribed 
rectangle. 

3.  Find  the  area  bounded  by  the  two  axes  and  the  curve  y  =  cos  x, 
in  the  first  quadrant. 

4.  Find  the  maxima  and  minima,  and  the  points  of  inflexion  (if  any 
^xist)  on  each  of  the  following  curves  : 

(a)  2/  =  sinx.  (d)   y  =  xsinx.  (ff)  y  =  e-*s\nx. 

(b)  y-cosx.  (e)   y  =  I  +  sin 2 x.  (h)  y  =  e-^s'mx. 

(c)  2/  =  tanx.  (/)  y  =  sin  x  +  cos  x.        (i)    y  =  cos(2x  +  w/6). 

5.  Find  the  derivative  of  each  of  the  following  pairs  of  functions, 
and  draw  conclusions  concerning  the  functions  : 

(a)  sinx  and  cos(7r/2  —  x).  (d)   sin  2  x  and  2  sin  x  cos  x. 

(b)  sin- X  and  I  —  cos^  x.  (e)    cos  2  x  and  2  cos^  x. 

(c)  cos  X  and  cos  ( —  x) .  (/)  tan'^  x  and  sec^  x. 


154  TRANSCENDENTAL  FUNCTIONS         [VI,  §  87 

6.  Integrate  the  following  expressions  ;  in  case  the  limits  are  stated, 
evaluate  the  integrals,  and  represent  them  graphically  as  areas  : 

(a)  y    sin  X  da;.  (c)  f"    sec'^xdx.       (e)     \  cos  {S  t +ir/ 6)  dt. 

(6)  I    '^    cosxdx.  (d)  \sin2xdx.  (/)    i  ta.n  t sec tdt. 

(g)  jj  (I  +  sin  x)  dx.  Uj)     (cos'^xdx. 

(h)  \{cosx  +  3sm2x)dx.  [hint.     2  cos^  a;  =  1  +  cos  2  x. 

(i)  i  (cos2a;  —  l)c?x.  (^•)    ( ''    sin^xdx. 

7.  Find  the  derivative  of  sin  x  directly  by  shovying  that 

sin  (x  +  Ax)  —  sin  X  =  sin  x  (cos  Ax  —  1)  +  cos  x  •  sin  Ax 
and  remarking  that 

lim  [(cos Ax—  1)-;-  Ax]=  0  and  lim  [(sin  Ax)-=-  Ax]  =  1. 
[See  §  13,  p.  19;  Ex.  8,  List  V;  and  §  96.] 

8.  Find  the  derivative  of  cos  x  directly  as  in  Ex.  7. 

9.  Find  the  derivatives  of  the  two  functions 

(a)  vers  X  =  1  —  cos X.  (6)  exsec  x  =  sec x  —  1. 

10.  Differentiate  each  of  the  ansioers  in  the  list  of  formulas.  Tables, 
IV,  Ea,  Ej.     What  should  the  result  of  your  differentiation  be  ? 

[The  teacher  will  indicate  which  formulas  should  be  thus  tested.] 

11.  Find  the  speed  of  a  moving  particle  whose  motion  is  given  in 
terms  of  the  time  t  by  one  of  the  pairs  of  parameter  equations  which 
follow  ;  and  find  the  path  in  each  case : 

'  X  =  2  cos  3  ^  f  X  =  sin «  +  cos  «. 


(")    [?/  =  2sin3f.  (^)    ly  =  sin«. 

'  X  =  2  cos  4it.  f  X  =  sec  t. 


2/  =  3  sin  4  «.  ^^^^    [  ?/  =  tan  i 

12.  A  flywheel  5  ft.  in  diameter  makes  1  revolution  per  second. 
Find  the  horizontal  and  the  vertical  speed  of  a  point  on  its  rim  1  ft.  above 
the  center. 

13.  A  point  on  the  rim  of  a  flywheel  of  radius  10  ft.  which  is  6  ft. 
above  the  center  has  a  horizontal  speed  20  ft.  per  second.  Find  the 
angular  speed,  and  the  total  linear  speed  of  a  point  on  the  rim. 


VI,  §  88] 


SIMPLE   H.\RMOXIC   MOTION 


155 


14.  The  cycloid  (Tables,  III,  Gi)  is  defined  by  the  equations 

sc  =  a  («  -  sin  <),  y  -  a(l  —  cost). 
Find  the  horizontal  and  the  vertical  speeds  if  t  represents  the  time  in  the 
motion  of  a  particle  for  which  these  equations  hold.     Find  the  total 
speed  ;  the  tangential  acceleration.    Find   the  values  of  each  of  these 
quantities  when  t  =  7r/4. 

15.  Find  the  area  of  one  arch  of  the  cycloid.     [See  Ex.  6,  (j)  ] 

16.  Show  that  the  differential  of  the  arc,  ds,  of  the  cycloid  is 

ds  =  aV2  —  2costdt  =  2asin((/2). 
Hence  find  the  length  of  one  arch  of  the  cycloid. 

88.    Simple  Harmonic  Motion.     If,  as  in  §  85,  a  point  3/ 
moves  with  constant  speed  in  a  circular  path,  the  projection 
Pof  that  point  on  any  straight  line  is 
said  to  be  in  simple  harmonic  motion. 

Let  the  circle  have  a  radius  a ;  let 
the  constant  speed  be  v ;  and  let  the 
straight  line  be  taken  as  the  cc-axis. 
We  may  suppose  the  center  of  the 
circle  lies  on  the  straight  line,  since 
the  projection  of  the  moving  point  on 
either  of  two  parallel  straight  lines 
has  the  same  motion.  Let  the  center 
0  of  the  circle  be  the  origin.  Then  we  have 
(1)  X  =  OP=a  cos  0,    or    x  =  a  cos  (s/a), 

where  s  =  arc  AM,  since  0  =  s/a.  Moreover,  since  the  speed  v 
is  constant,  v  =  s/T,  if  T  is  the  time  since  M  was  at  ^;  or 
v  =  s/{t—  to)  if  t  is  measured  from  any  instant  whatever,  and 
^0  is  the  value  of  t  when  M  is  at  A.     We  have  therefore 


g(.-.)]=« 


(2)  x  =  a  cos  -  = 

where  k  =  v/a,  and  e  =  —  7it^  =  —  Wo/a. 

From  (2),  the  speed  clx/cU  of  P  along  BA  is 


cos[A^<  +  e] 


(3) 


dx 


d\acoMlct  +  .)-]  ^  _  ^^  gi^ ^^.^  ^  ^^^ 


dt 


156  TRANSCENDENTAL  FUNCTIONS         [VI,  §  88 

and  the  acceleration  of  P  is 

(4)  ^V=^  =  -«^'cos(A;«  +  e)=-A;2.a;, 
or, 

(5)  jV-^  =  ^'-^i«  =  -^'; 

that  is,  the  acceleration  of  x  divided  by  x,  is  a  negative  constant, 
—  !<?.  We  shall  see  that  much  of  the  importance  of  simple 
harmonic  motion  arises  from  this  fact. 

It  is  important  to  notice  that  (2)  may  be  written  in  the 
form 

x  =  a  cos  (kt  +  e)  =  a  [cos  e  cos  M  —  sin  e  sin  kf], 
or 

(6)  a-  =  A  sin  kt  +  B  cos  kt, 

where  A=  —  a  sin  e  and  B—  +a  cos  e  are  both  constants. 
The  form  (6)  may  be  used  to  derive  (5)  directly. 

The  simplest  forms  of  the  equation  (6)  result  when  ^■  =  l 
and  either  A  —  {)  and  £  =  1,  or  ^  =  1  and  jB  =  0 : 
,        I  .T  =  sin  ^ ;  if  A;  =  1,  ^  =  1,  B  =  0,  i.e.  a  =  1,  e  =  3  7r/2. 
^'  ^     L-  =  cos  i ;  if  A;  =  1,  A  =  0,  B  =  l,  i.e.  a  =  l,e  =  0. 
The  formulas  (2)  and  (6)  are  general  formulas  for  simple  har- 
monic motion  ;  (7)  represents  two  especially  simple  cases. 

89.  Relative  Acceleration.  The  ratio  of  d^x/dt-  to  x  found 
above  is  the  relative  acceleration  of  x. 

In  Ex.  8,  p.  149,  we  saw  tliat  the  function  a;  =  ae*' ''"''  gave 
d?x/df  —  Ic^x,  or  {d-x/dt^)  -=-  .t  =  A^^  that  is,  the  relative  accelera- 
tion of  aj  is  a  positive  constant,  A;^. 

In  §  88,  we  saw  that  a  simple  harmonic  motion,  represented 
by  (2)  or  (6)  of  §  88,  gives  d^x/df^x=  -k^;  that  is,  the  rela- 
tive acceleration  is  a  negative  constant,  —  k'\  It  will  appear 
later  (see  §  187)  that  these  are  the  only  functions  for  which  the 
relative  acceleration  is  a  constant  different  from  zero. 


i 


VI,  §  91]  SIMPLE   HARMONIC   MOTION  157 

90.  Vibration.  riie  importance  of  simple  harmonic  motion,  based 
on  its  property  (5)  of  §  88,  is  evident  in  vibrating  bodies,  such  as  vibrat- 
ing cords  or  v?ires,  the  prongs  of  a  tuning  fork,  the  atoms  of  water  in  a 
wave,  a  weight  suspended  by  a  spring. 

In  all  such  cases,  it  is  natural  to  suppose  that  the  force  which  tends 
to  restore  the  vibrating  particle  to  its  central  position  increases  with  the 
distance  from  that  central  position,  and  is  proportional  to  that  distance. 
(Compare  Hooke's  law  in  Phj^sics.) 

It  is  a  standard  law  of  physics,  equivalent  to  Newton's  second  law  of 
motion,  that  the  acceleration  of  any  particle  is  proportional  to  the  force 
acting  upon  it.     (See  §  i>2,  p.  82.) 

In  the  case  of  vibration,  therefore,  the  acceleration,  being  proportional 
to  the  force,  is  proportional  to  the  distance,  x,  from  the  central  position ; 
it  follows  that,  in  ordinary  vibrations,  the  relative  acceleration  is  a  negative 
constant,  —  negative,  because  the  acceleration  is  opposite  to  the  positive 
direction  of  motion.  For  this  reason,  each  particle  of  a  vibrating  body 
is  supposed  to  have  a  simple  harmonic  motion,  unless  disturbing  causes, 
such  as  air  friction,  enter  to  change  the  result.  Neglecting  such  frictional 
effects  temporarily,  the  distance  x  from  the  central  position  is,  as  in  §  88, 

X  —  a  cos  (kt  +  e)  =  A  sin  kt  +  B  cos  kt, 
where  t  denotes  the  time  measured,  from  a  starting  time  to  seconds  before 
the  particle  is  at  x  —  a,  and  where  e  =  —  tok.     Moreover,  from  §  88  and 
also  from  what  precedes,* 

The  quantity  a  is  called  the  amplitude,  2  tr/k  is  called  the  period,  and 
to  =  —  e/k  is  called  the  phase,  of  the  vibration. 

91.  Waves.  Another  important  application  of  S.  H.  M.  is  in  the 
treatment  of  wave  motions.  Thus  the  form  of  a  simple  vibration  of  a 
stretched  cord  or  wire  is  assumed  to  be 

y  =  asiuyTT, , 

*  Electric  vibrations  follow  this  same  law  if  the  resistance  is  negligible. 
If  V  represents  the  electromotive  force  in  volts,  d^v/dt'^=  —k'h,  where  k  is  a 
constant.  The  sudden  discharge  of  an  electric  condenser  by  a  good  conductor 
would  give  such  an  electric  vibration.  But  tlie  etTect  of  the  electric  resist- 
ance (which  corresponds  to  the  friction  in  mechanical  vibrations)  is  very 
marked,  and  the  vibrations  die  out  with  extreme  rapidity. 


158 


TRANSCENDENTAL  FUNCTIONS         [VI,  §  91 


where  I  is  the  total  length  of  the  cord  between  the  fixed  ends  and  n  is 
the  number  of  arches  in  the  wave. 


y 

^ 

V 

^ 

v^ 

X 

^ 

^ 

^ 

, 
^ 

S^ 

x 

*> 

J 

_ 

- 

_ 

_ 

_j 

y  =  sin^-7rfor  ?=5?i;  i.e.  y  =  sin -— . 
I  5 

Fig.  39. 

A  compound  vibration  of  such  a  stretched  cord  is  thought  of  as  made 
up  by  combining  several  such  simple  vibrations  simultaneously : 


2/  =  ai  sin  ^?^  IT  +  ffla  sin  ^^  TT  + 


+  ap  sin  "P^  TT. 


An  alternating  electric  current  varies  with  the  time  in  a  similar  man- 
ner ;  for  a  simple  alternating  current, 

C  =  a  sin  ^t, 

where  C  is  the  current  in  amperes  and  t  is  the  time  measured  in  seconds 
from  a  time  when  C  =  0  ;  or  the  sum  of  several  such  terms  for  a  com- 
pound current. 

In  general,  a  sum  of  several  simple  harmonic  terms  : 

a\  sin  {kxt  -\-  ei)  -|-  as  sin  (^•2^  +  62)+  •■•  4-  «;,  sin  {kj,t  +  e^) 

is  called  a  compound  harmonic  function.     See  Tables^  III,  F. 


EXERCISES  XXXrV.  — SIMPLE  HARMONIC  MOTION  —  VIBRATIONS 

1.  Find  the  speed  and  the  acceleration  of  a  particle  whose  displace- 
ment X  has  one  of  the  following  values  ;  compare  the  acceleration  with 
the  original  expression  for  the  displacement  : 

(fl)  «  =  sin2«.  (e)  a;  =  sin2  « -|- 0.15sin6«. 

(6)  a;  =  sin  (//2  -  7r/4).  (/)  x  =sini;- ^sin3«-F  ^sin  5«. 

(c)  a;  =  sin «  —  I  sin  2 1.  (g)  x  =  a  sin  {kt  +  e) . 

(d)  r  =  cos « -f  i  cos  3 1.  (h)  x-  A  cos  kt  +  B  sin  kt. 

2.  Determine  the  angular  acceleration  of  a  hair  spring  if  it  vibrates 
according  to  the  law  e  =  .2  sin  10  irt  ;  what  is  the  amplitude  of  one  vibra- 
tion, the  period  and  the  extreme  value  of  the  acceleration  ? 


VI,  §  91]  SIMPLE  HARMONIC   MOTION  159 

3.  Show  that  each  of  the  following  functions  satisfies  an  e(iuation  of 
the  form  d-u/dt-  +  k-u  =  0  or  d^u/dt-  —  k'-ii  =  0  ;  in  each  case  determine 
the  value  of  k  : 

(ffl)  ti  =  lOsmSt.  (/)  ?(  =  5cos(</2  -  7r/12). 

(6)  n  =  0.7  cos  13  f.  (g)    u  =  12cos4  t  —  5sm4t. 

(c)  u  =  !ie'^.  (A)   ?f  =  3sin5«  +  4  cos5^ 

(d)  u  =  20  e-2«.  (0    ti  =  Ci  sin  3  «  +  Co  cos  3  «. 

(e)  u  =  sin  (5  f  +  jr/3).  (j)    m  =  Cic^  +  Cafi-*'. 

4.  Show  that  the  function  u  =  A  sin  kt  +  -B  cos  if  always  satisfies  the 
equation  d^u/dfi  4-  k!^u  =  0  for  any  values  of  A  and  B.  Check  by  substi- 
tuting various  positive  and  negative  values  for  k,  A,  B. 

5.  Show  that  u  =  Ae^  +  jBe-**  always  satisfies  the  equation 

d-u/dt-  —  k-^i  =  0. 

6.  Substitute  u  =  e"^  in  the  equation  d-u/dx^  —  -iu,  and  show  that  e"" 
is  a  solution  if,  and  only  if,  m^  —  4.  Show  that  u  =  Ae^  +  Be-^  is  a 
solution  for  all  values  of  A  and  B. 

7.  By  the  methods  of  Exs.  4-G,  write  down  by  inspection  as  general 
a  solution  as  possible  for  each  of  the  following  equations  : 

(«)  ^  =  _x.  (c)    ^  =  -lx.  (e)    ^=16x. 

^  '   df^  ^  ^    dfi         4  ^  ^    df^ 

(6)    ^  =  -4x.  (d)   ^  =  9x.  (/)   ^=l2z. 

at-  dt'-  dt^ 

[Note.  Any  equation  which  contains  derivatives  is  called  a  differen- 
tial equation.  Many  simple  ones  have  been  used.  It  is  shown  in 
Chapter  X  that  the  solutions  found  in  Exs.  4-G  are  the  most  general 
solutions.] 

8.  The  differential  equation  of  falling  bodies  is  d^s/dt-  =—  (j;  show 
that  S--  gt-/2  +  Cit  +  Co.  Find  Ci  and  C-  if  s  =  0 and  the  speed  v  =  0 
when  t  =  0  ;  if  s  =  0  and  v  =  100,  when  t  —  0. 

9.  The  differential  equation  of  a  certain  vibrating  body  is  d-s/dt'-  =  —  s  ; 
show  that  s  —  As'nit -{■  Bcost ;  find  A  and  B  il  s  —  0  and  the  speed 
«  =  10  when  t  =  0;  if  s  =  2  and  v  =  0  when  t  =  0. 

10.  A  flywheel  6  ft.  in  diameter  revolves  with  a  uniform  speed  of 
30  R.  P.  M.  Write  the  differential  equation  of  the  projection  on  the  floor 
of  a  point  on  the  rim. 

11.  A  horizontal  slider  *S'  attached  by  an  exceedingly  long  connecting 
rod  SP  to  a  pivot  P  on  a  wheel  whose  center  is  0,  is  forced  to  move 


160  TRANSCENDENTAL  FUNCTIONS         [VI,  §  91 

approximately  as  the  projection  of  P  on  the  floor.  If  the  wheel  rotates 
uniformly,  show  that  the  distance  s  of  the  slider  from  its  central  position 
is  approximately  s  =  a  sin  kt,  where  a  =  OP  (the  "  crank''^ ),  and  ^■  =  2  ttm, 
where  the  wheel  revolves  n  times  per  second,  and  OP  is  vertical  when 
t  =  0. 

12.  If  OP  in  Ex.  11  makes  an  angle  e  with  the  vertical  when  t  =  0, 
show  that  s  —  a  sin  {kt  +  e). 

[Note,  a  is  called  the  amplitude,  n  =  k/2  ir  the  frequency,  and  e  the 
phase  (or  phase-angle)  of  the  S.  H.  M.  of  the  slider.] 

13.  When  an  electrical  condenser  discharges  through  a  negligible  re- 

^■2  (J 

sistance  the  current  C  follows  the  law =  —  a^C,  where  a  is  a  constant. 

d(- 

Express  the  current  in  terms  of  the  time.     When  a  =  1000,  what  is  the 

frequency  (number  of  alternations)  per  second  ? 

14.  Any  ordinary  alternating  electric  current  varies  in  intensity 
according  to  the  law  C  =  a  sin  kt ;  find  the  maximum  current  and  the 
time-rate  of  change  of  the  current. 

15.  Show  that  two  terms  of  the  form  (a  cos  kt  +  b  sin  kt)  and 
(A  cos  kt  +  B  sin  kt)  combine  into  one  term  of  the  same  general  type. 

16.  Show  that  two  terms  of  the  form  ai  sin  (kt  -\-  ei)  and  aosin  (Ari  +  eo) 
combine  into  one  term  of  the  type  mentioned  in  Ex.  15. 

17.  When  a  pendulum  of  length  I  swings  through  a  small  angle  6,  its 

motion  is  very  closely  represented  by  the  equation  —  =  —  ^0,  I  being 

d«-  I 

in  feet,  6  in  radians,  t  in  seconds.     Show  that  d  =  Ci  sin  kt  +  C-2  cos  kt, 

where  k  =  y/gJT.     Find  Ci  and  Co  if  »  =  a  and  the  angular  speed  w-0 

when  t  =  0;  and  find  the  time  required  for  one  full  swing. 

18.  A  needle  is  suspended  in  a  horizontal  position  by  a  torsion  fila- 
ment. When  the  needle  is  turned  through  a  small  angle  from  its  position 
of  equilibrium,  the  torsional  restoring  force  produces  an  angular  accelera- 
tion proportional  to  the  angular  displacement.  Neglecting  resistances, 
what  will  be  the  nature  of  the  motion. 

92.   Damped  vibrations.    The  curve 
(1)  y  =  e-' 

has  been  drawn  in  several  examples;  its  relative  rate  of  increase, 
dy/dt  -r-y,  is  —  1.  Hence  the  relative  rate  of  increase  of  y  is  —  1  ;  or, 
the  relative  rate  of  decrease  oi  y  is  4-1. 


VI,  §  92] 


DAMPED  VIBRATIONS 


161 


If  a  vibration  would  follow  the  law 

(2)  y=asinkt 
when  not  aifected  by  friction,  the  formula 

(3)  y  —  ae~f  sin  kt, 

in  which  a  is  replaced  by  oe-',   expresses   a  corresponding  damped 
vibration,  in  which  the  total  amount  of  displacement  >j  is  equal  at  any 


Fig.  40  « 

instant  t  to  the  value  given  by  a  formula  like  (2)  in  which  a  diminishes, 
the  relative  rate  of  decrease  in  a  being  +  1.  The  curve  is  shown  in 
Fig.  40  (c)  ;  it  may  be  obtained  by  drawing  the  ordinates  in  (1)  multi- 
plied by  the  corresponding  ordinates  in  (2). 


^ 

— 

— 

— 

— 

y 

y-- 

-- 1 

s 

ll, 

t 

N 

/ 

r 

\ 

y 

r 

N 

/ 

\ 

t 

\ 

y 

\ 

y 

V 

y 

\ 

-J 

\                                       7/-     fip't   sin  Kl 

-\~~---------------- 

:]-::S;:::;::::::7 

::;==:=====:===:== 

Fig.  40^ 


162  TRANSCENDENTAL   FUNCTIONS         [VI,  §  92 

Likewise, 

(4)  y  =  ae-*'  sin  {kt  +  e) 
is  a  damped  vibration,  wliich  may  be  written 

(5)  y  =  A  sin  {kt  +  e) ,    wliere  A  =  ae-K 

Here  A  is  a  variable  decreasing  amplitude,  whose  relative  rate  of 
decrease  is  —  dA/dx  -i-A—b;  that  is,  the  relative  rate  of  decrease  of  A 
is  constant.* 

The  successive  derivatives  of  y,  by  (4),  are: 

^  =  rte-6'  [-  b  sin(A-«  +  e)-\-k  cos  {kt  +  e)], 
^  =  ae-"  [(6-  -  ^•■')  sin  (A.-«  +  e)  -  2  hk  cos  (^•«  4  e)], 
whence  it  follows  that 

Equations  which  contain  derivatives  are  called  differential  equa- 
tions ;  thus  (6)  is  the  fundamental  differential  equation  for  damped 
vibrations. 

EXERCISES   XXXV.  — DAMPED   VIBRATIONS 

1.  Each  of  the  following  equations  represents  a  damped  harmonic 
vibration  ;  find  the  speed  and  the  acceleration  in  each  case  ;  and  write 
an  equation  connecting  the  acceleration,  the  speed,  and  the  value  of  y. 
Draw  the  graph  of  each  equation. 

{a)  y  -e-*sva.2t.  (d)   y  =  2 e-io« cos 5 «. 

(6)  y  =  e-2<  cos  4  t.  (e)    ?/  =  2  e-5«  sin  (2  t  +  v/S). 

(c)   y  =  5e-^sin"t.  (/)  ?/ =  4  e-i«cos  (3«-5  7r/12). 

*  In  common  language,  this  is  often  expressed  by  saying  that  "  the  vibra- 
tion dies  away  regularly,"  or  "fades  out  uniformly."  The  fact  that  the 
relative  rate  of  decrease  of  A  is  constant  is  the  fundamental  assumption. 

t  The  equation  (6)  is  often  obtained  directly  and  solved  to  obtain  (4)  as  in 
Chapter  X  ;  the  assumptions  made  in  this  work  are  equivalent  to  the  assump- 
tion just  mentioned,  —  that  the  relative  rate  of  decrease  of  A  is  constant; 
this  assumption  is  really  the  fundamental  one,  and  its  reasonableness  is  the 
real  justification  of  the  assumptions  made  when  (6)  is  obtained  first.  The 
term  in  dy/dt,  or  v,  proportional  to  the  velocity,  occurs  only  when  "  damping" 
Cor  friction)  is  considered.    A  similar  equation  governs  electric  vibrations. 


VI,  §  93]    INVERSE  TRIGONOMETRIC   FUNCTIONS     163 

2.  The  factor  e-''  produces  a  more  rapid  damping  effect.  Draw 
y  —  g-'^sini,  and  compare  it  with  y  —  e-'  sin  t.  Find  the  speed  and  the 
acceleration  in  each  case. 

3.  The  factor  (1  +  «2)-i  (Example  2,  p.  1G5)  produces  an  effect  similar 
to  that  of  the  factor  e-«'.  Draw  y  —{\-\- 1-)-'^  sin  t ;  find  the  speed  and 
the  acceleration. 

4.  Show  that  y  =  e~''sin  t  satisfies  the  equation 

d-y/dC^  +  4  t  (dy/dt)  +  (3  +  if^)y  =  0. 

5.  Draw  the  curve  y  =  sech  t-s'mt;  compare  it  with  y  =  e-''sin  t ; 
find  the  speed  and  the  acceleration. 

93.   Inverse  Trigonometric  Functions.     Since  the  equations  * 

(1)  y  =  sin  X,    x  =  sm~^y  (=arc  sin  y) 
are  equivalent,  it  follows  that 

rXVIl  <?  sin~^  y  _dx  _      1      _     1     _        1 

fly  cly     dy/dx     cos  x     Vl  -  y^ 

a  formula  which  may  be  written  in  other  letters  when  con- 
venient.    It  is  evident  that  the  radical  should  have  the  same 
sign  as  cos  x,  i.e.  +  when  x  is  in  the  1st  or  4th  quadrants ; 
—  when  a;  is  in  the  2d  or  3d  quadrants. 
Likewise,  from 

(2)  y  =  cosx,  or  x  =  c,os~^y  (=  arc  cos?/). 

[XVIII         dcm-^y_dx_     1      _      1       _      — 1 

dy  dy     dy/dx     —  sin  x     VI  -  y- 

where  the  sign  —  applies  when  sin  x  is  positive,  i.e.  for  values 
of  X  in  the  1st  and  2d  quadrants. 

In  like  manner,  the  student  may  show  that 

[XVIII]  ^^7"'y  =  -^  (all  values  of  y)  ; 

dy  1  +  2/2  V  ^^' 

[XIX]  a^:^  ^  ^^  (all  values  of  y)  ; 

dy  1+y-  ^  '^^ ' 

*  The  symbols  sin-i  y  and  arc  sin  y  will  both  be  ased  to  denote  the  a7iffle 
wJiose  sine  is  y.  In  writing  such  formulas  as  those  on  this  page,  the  notation 
sin-l  is  the  shorter. 


164 

TRANSCENDENTAL   FUNCTIONS         [VI,  §  93 

[XX] 

d sec-i  y            1 

(x  =  sec-i  y  in  1st  or  3d  quadrant)  ; 

<^y        yVy^-1 

[XXI] 

d  csc-i  y         -  1 

(x  =  csc-i  y  in  1st  or  3d  quadrant) ; 

cly          yVy'-l 

[XXII] 

d  vers"!//             1 

(,r  =  vers"^?/  in  1st  or  2d  quadrant) ; 

V  2  y  -  2/ 


94.  Integrals  of  Irrational  Functions.  The  preceding  for- 
mulas are  of  little  value  as  direct  differentiation  formulas. 
The  reversed  differentiations  derived  from  them  are  very  im- 
portant, because  they  show  how  to  obtain  the  integrals  of 
certain  simple  irrational  functions. 

Thus,  using  the  letter  x  in  place  of  y,  the  formulas  [XVI], 
[XVIII],  [XX],  and  [XXII]  become 

r  v.rw-1         r     due  '     ,      .  n      ■        d  siu~'  X  1 

[XVI]i        j  =sm-iic  +  (7,  since  - 

«/  Vl  -  ai^ 


[XVIII], 


dx  Vl  -  x^ 

f-^      =  tan-  X  +  a  since  ^^^  =  -J- , 

J  1  +  052  '  dx  1  +  a^ 

dx  _!./-/•        d  sec~^  X  1 


/; 


[XX] j        i ---—  =  sec~^  x-{-G,  since 


[XXII]; 


Va^-l  dx  x^x'-l 

dx  ^^        ri     •        d  vers"^  x  1 


/dx  ^  n       • 

— ^—^—  =  vers"*  X  +  C,  since 
a/2.7;  _  cri^ 


^/2x-x'  dx  V2: 


where  C  in  each  case  denotes  an  arbitrary  constant.  Since 
sin~^  a:-|-cos~^  a;=7r/2,  the  student  may  show  that  [XVII]  leads 
to  the  same  result  as  [XVI]. 

95.  Illustrative  Examples.  A  few  illustrative  examples  will 
be  given  here ;  in  Chapter  VII  many  other  integrals  of  irra- 
tional functions  are  found  by  means  of  those  just  written. 


VI,  §  95]    INVERSE  TRIGONOMETRIC  FUNCTIONS      165 

Example  1.     Given  y  —  sin-i  (.r-),  to  find  dy/dx. 
Method   1.      Set    x- =  u.      Then    dy/dx  =  (dy/du)  (du/dx).      Since 
dy/du  =  d  sin-i  2</c?w  =  1/  Vl  -  M'^   and  dw/dx  =  d(x-)/dx  =  2x, 
dy/dx  =  (l/Vl  -M-!)  2  X  =  2  x/Vl  -  x*. 
Method  2.     (Zi/  =  d sin-i(x-2)  =  [1/Vl  -  (x-iy]d(x:^)  =  (2  x/ Vn^^)dx. 
Notice  that  the  resulting  integral  formula  may  be  written 
C_dn_ ^  r  _2^^  ^  ^j„.,  ^^  ^  ^  ^  ^j,^_,(^,^ _^  ^_ 
•^  Vl  -  ?<2      ^  Vl  -  a,-* 

Example  2.     To  find  the  area  under  the  curve  ?/  =  1/(1  +  x^)  from  the 
point  where  x  =  0  to  the  point  where  x  =  1. 
Since  A  =  iydx,  we  have 

^l"^^  =  C^^  -^— ^  dx  =  tan-i  xl"^^  =  7r/4  -  0  =  7r/4. 

T7ie  /ac«  fAaf  toe  are  iising  radian  measure  for  angles  appears  very 
prominently  here.  Draw  the  curve  (by  first  drawing  ?/  =  1  +  a--)  on  a 
large  scale  on  millimeter  paper  and  actually  count  the  small  squares  as  a 
check  on  this  result. 

EXERCISES  XXXVI.— INVERSE  TRIGONOMETRIC  FUNCTIONS 

1.    Differentiate  each  of  the  following  functions  : 

(a)  sin-ix3.  (e)    sin"i  (1  —  x-).  (i)    logcos-^x. 

(b)  cos-i(l  +  x).  (/)  xcos-ix.  (i)  siu-i(xe»)._ 

(c)  sin-i(l/a;).  (fir)    tan-i(l/x2).  (k)  x2tan-i2v'x. 

(d)  tan-i(2a:).  (A)    e^sin-ix.  (0    sec-i(a;2  +  4x). 

2.  Given  versx  =  1  —  cosx,  show  that  the  derivative  of  vers"^*  is 
l/V2x-x^. 

3.  Given  exsec  x  =  secx  —  1,  find  the  derivative  of  exsecx. 

4.  Integrate  the  following  functions  ;  in  case  limits  are  stated,  evalu- 
ate the  integral  : 


(«)  ^^\J^^.  id)  c 

•^"      1  +  x^  -^1 


(6)     ('  -^—                    (e)     f^-^-      [Set«  =  2x.] 
(c)     r^'-i^.  (/)    f     Jl •      [Set«  =  2x.] 

J-l    1  +  X2  J    VI  -4x2 


166  TRANSCENDENTAL  FUNCTIONS         [VI,  §  95 

5.  Find  the  areas  between  the  x-axis  and  each  of  the  following  curves, 
between  the  limits  stated  : 

(rt)   j/'-i  =  1  +  or^y^- ;  x  =  0  to  x  =  1/2  ;  x  =  -  1/2  to  X  =  +  1/2. 
(6)  2/  +  x2?/  =  1  ;  X  =  0  to  X  =  1 ;  X  =  0  to  X  =  a. 
(c)    2/2  =  1+4  xhf-  ;  X  =  0  to  X  =  1/4  ;  x  =  -  1/4  to  x  =  +  1/4. 
(d)4x2?/  +  ?/  +  l=0;x  =  ltox  =  2;x  =  -ltox  =  +l. 

6.  Integrate  after  making  the  change  of  letters  m  =  1  —  x  : 

J  Vl-(l-xr         '  '  Jl+Cl-:*:)--^  '^V2^^^ 

7.  Show  by  differentiation  that  sin-i  x  and  —  cos~i  x  differ  by  a  constant. 
Find  the  value  of  that  constant  by  elementary  trigonometry. 

8.  Show  that  sin-^x  and  tan-i  [x/(l  —  x2)V2]  differ  at  most  by  a 
constant.     By  trigonometry,  show  that  the  two  functions  are  equal. 

9.  Show  that  sin-i  (1  —  x)  and  vers~i  x  differ  by  a  constant.  Show 
that  cos-i(l  —  x)  =  vers~i  x. 

10.  Show  that  the  derivative  of  tan-i  [(e^  -  e-^)/2]  is  2/{e^  +  e-') . 
[Note.     The  function  tan-i[(e^  —  e-=^)/2],  or  tan-i  (sinhx),  is  called 

the  Gudermannian  of  x  and  is  denoted  hj  gd x  :  gdx  =  tan-i  (sinhx). 
It  follows  from  this  exercise  that  dgd  x/dx  =  sechx.] 

11.  From  the  fact  that  d(sinh*x)  =  coshxfZx,  show  that  the  derivative 
of  the  inverse  hyperbolic  sine  (x  =  sinh-i  t(  if  m  =  sinh  x)  is  given  by 
the  equation  d(sinh-i  m)  =  [1/(1  +  u^y/^^du.     [See  Ex.  4,  p.  140.] 

12.  Showthat  dcosh-iM  =  ±[l/(M2-  1)1/2](Zm. 

13.  Show  that  d  tanh-i  t<  =[1/(1  —  ■u-)}du. 

96.  Polar  Coordinates.  Equations  of  curves  in  polar  coordi- 
nates frequently  involve  trigonometric  functions.  Given  a 
curve  C  whose  equation  in  polar  coordinates  is 

(1)  P=/W- 

If  PB  is  the  arc  of  the  circle  about  0  with  radius  p  =  OP, 
then  BQ  =  Ap;  while  arc  PB  =  p  ■  Z  POB  =  pM.     Hence 

^  ^        Ad     ^arc  PB      ^  '  AQ  '  arc  PB     PA 


VI,  §  96]  POLAR  COORDINATES  167 

Since  PA  =  p  sin  A^,  FA/a,rc  PB=  sin  \B/\d  approaches  1 
by  §  13,  p.  19.     Since  OA  =  p  cos  A^, 

AQ  _  p  +  Ap  —  p  cos  A^  ^  -,  cos  Ag  —  1     A^ 

BQ~  Ap  ~         ^         A^  ■  Ap' 


which,  approaches  1  since 

lim  cos  Ag  - 1  ^  J.  j^^  cos  (0  +  ^0)  -  cos  0 

A0H)  A^  A0^  Ad 


It  follows  that 


(3)        -^'^-  =  1  ini  ^  =  p  lim  ^  =  p  lim  ctn  (^  =  p  ctn  ih, 

where  <^  =Z  DQS  between  the  secant  (S)  and  the  radius  vector 
OD  &nd  where  i(/  =  ZBPT  between  the  tangent  (T)  and  the 
radius  vector  OR.  The  geometrical  meaning  of  the  derivative 
clp/rW  in  polar  coordinates  is  p  ctn  if/. 

Dividing  both  sides  of  this  equation  by  p: 

dlogp^^  ^..^^ 

dO         dd     ^  "^  '^' 

where  ;^(=90°  — i/')  is  the  angle  between  the  curve  and  circle 
about  0  through  the  point  P:  the  relative  rate  of  increase  r,  of 
p  with  respect  to  0  is  the  tangent  of  the  angle  ;^.     (See  §  83,  p.  147.) 


168  TRANSCENDENTAL   FUNCTIONS         [VI,  §  96 

The  polar  coordinate  diagrams  can  therefore  be  used  very 
effectively  to  represent  quantities  whose  relative  rates  of  change 
are  important.  Eor  example,  curves  showing  the  growth  of 
population  of  cities  or  countries  may  well  be  drawn  on  polar 
coordinate  paper,  the  time  being  represented  by  the  angle  6. 

The  angle  a  between  the  tangent  ( T)  and  the  .T-axis  can  be 
found  after  \p  has  been  found  by  means  of  the  relation  a  =  d  -\-\p. 

Example  1.  Given  the  curve  p  =  e^,  find  dp/dd  and  Vr  =  tan  x 
=  dp/de  ^  p  =  d  (log  p)/dd.     (See  Tables,  III,  M.) 

Since  p  =  e^,  dp/dd  —  e^,  and  r  =  dp/dd  -^  p  =  1.  Hence  x  =  tan-i  1 
=  Tr/4  =  45'^  ;  tliis  curve  cuts  every  radius  vector  at  the  fixed  angle  45°. 

Physical  experiments  (see  §123,)  in  which  it  is  suspected 
that  the  quantities  measured  follow  a  compound  interest  laiv 
can  be  tested  by  plotting  in  polar  coordinates ;  the  angle  i// 
(and  therefore  also  x)  should  be  constant  (see  Ex.  4  below). 

Example  2.     Given  p  =  sin  2  0,  find  ^  at  the  point  where  6  —  ir/S. 

ctn  i/'  =  ^  -J-  p  =  2  cos  2  ^  H-  sin  2  6>  =  2  ctn  2  ^. 
^     de 

Hence  ctn  i//  =  2  when  e  =  v/8,  whence  ^|/  =  26°  34'. 


EXERCISES  XXXVII.— POLAR  COORDINATES 

1.  Plot  each  of  the  following  curves  in  polar  coordinates ;  find  the 
Talue  of  ctn  f  in  general,  and  the  value  of  \p  in  degrees  when  ^  =  0,  ir/6, 
7r/4,  7r/2,  tt. 

(/)  p  =  e.  (k)    p  =  sin  2  e. 

(g)   p  =  e^.  (?)    p  =  2  cos  3  e. 

(h)  p  =  1/e.  (m)  p  =  3  sin  (3  e  +  2  7r/3). 

(f )    p  =  e-«.  (n)    p  =  3  cos  (?  +  4  sin  d. 

(i)    p=e-'e,         (o)   p  =  2/(1- cos  ^). 

2.  Show  that  ctn  \p  is  constant  for  the  curve  p  =  ^•e«*. 

3.  Show  that  ctn  \{/  for  the  spiral  p  =  A;^  is  greater  than  ctn  \j/  ior  p  =  e^ 
when  e  <  1.  Hence  show  that  the  former  winds  up  more  rapidly  than 
the  latter,  as  p  ^  0. 

4.  Show  that  if  the  curves  p  =  e^^  are  supposed  drawn,  for  various 
values  of  a,  any  function  p  =/(^)  whose  relative  rate  of  change  (loga- 


(a) 

P- 

=  4  sin  0. 

(&) 

P- 

=  6  cos  ^  —  5. 

(<') 

P- 

=  3  +  4  cos  (9 

(d) 

P- 

=  tan  0. 

(e) 

P  - 

=  2  +  tan^  0. 

VI,  §  97] 


CURVATURE 


169 


rithmic  derivative)  is  variable  crosses  them  ;  shov?  that  the  new  curve 
moves  across  the  othei-s  away  from  the  origin  if  its  relative  rate  of 
change  is  increasing  as  0  increases. 

5.  Find  ctn  \p  for  each  of  the  following  curves  : 

(a)  p  =p/(\  -  e  cos  0)  (conic).  (c)    p  =  a(l  +  cos  0)  (cardioid). 

(6)  p-as.ecd±b  (conchoid).  (d)  p^  =  2  a-  cos  2  0  (lemniscate). 

6.  Find  the  value  of  tan  «  [Fig.  41]  in  terms  of  the  angles  0  and  f. 
Find  tan  a  for  each  of  the  curves  of  Ex.  1,  at  the  points  specified. 

97.  Curvature.  An  important  application  of  these  formulas 
consists  in  finding  a  more  accurate  measui-e  of  the  bending  of 
a  curve. 

The  Jlexion  (§  45,  p.  71), 

dm  _ 
dx      daf' 

is  a  crude  measure  of  the  bending ;  but  it  evidently  depends 
upon  the  choice  of  axes,  and  changes  when  the  axes  are  ro- 
tated, for  example. 

If   we   consider   the  rate   of 
change      of      the      inclination 
a  =  tan~^  w  with  respect  to  the 
length  of  arc  s,  that  is, 
A  a      da 


(1) 


7  Uiiio HI/ 


(2) 


lim 

A8=o  As      ds' 


it  is  evident   that  we    have    a 
measure  of  bending  which  does 

not  depend  on  the  choice  of  axes,  since  A«  and  As  are  the 
same,  even  though  the  axes  are  moved  about  arbitrarily,  or, 
indeed,  before  any  axes  are  drawn.  The  quantity  da/ds  is 
called  the  curvature  of  the  curve  at  the  point  P,  and  is  de- 
noted by  the  letter  K:  the  curvature  is  the  instantaneous  rate 
of  change  of  a  per  unit  length  of  arc. 

Since  «  =  tan~^m,  and  since  ds- —  dx- -^  di/  (§  62,  p.  107), 
we  have. 


170 


TRANSCENDENTAL   FUNCTIONS         [VI,  §  97 


da 

=  d  tan-^?n= 

1  +m 

-dm,   ds=Vl 

+  m' 

dx, 

where 

m  = 

dy/dx; 

hence  the  curvature  K  is 

(3) 

K 

da. 
-ds- 

--^^-dm 
1  +  m"" 

dm 
dx 

b 

V'l  +  m^  dx 

(1  +  m'f/' 

(1  + 

m2)3/2 

where  b  =  d^y/dx^  (=  flexion),  and  m  =  dy/dx (=s\o-pe).  It 
appears  therefore  that  the  flexion  b  when  multiplied  by  the 
corrective  factor  1/(1  +  m^f^^  gives  a  better  measure  of  the 
bending,  since  K  is  independent  of  the  choice  of  axes. 

The  reciprocal  of  K  grows  larger  as  the  curve  becomes 
flatter;  it  is  called  the  radius  of  curvature,  and  is  denoted 
by  the  letter  R: 

(4,)  it=±  =  ^  =  (l  +  OTt2)S/2 

^  ^  K     da  b 

It  should  be  noticed  that  this  concept  agrees  with  the  elementary  con- 
cept of  radius  in  the  case  of  a  circle,  since  As  =  /•  Aa  in  any  circle  of 

radius  r. 

Substituting  the  values  of  b  and 
m,  formulas  (3)  and  (4)  may  be 
written  in  the  forms 

d-y 
dx2 


K: 


Since  Vl  +  m-  = 
K^l/B^bcossa. 


dry 
dx' 
It  is  preferable,  however,  to  cal- 
culate m  and  b  first,  and  then  sub- 
stitute these  values  in  (3)  and  (4). 
sec  a  the  formulas  may  also  be  written  in  the  form 


VI,  §  97]  CURVATURE  171 

It  is  usual  to  consider  only  the  numerical  values  of  A'  that 
is  I  K  I,  without  regard  to  sign.  Since  K  and  b  have  the  same 
sign,  the  value  of  K  given  by  (3)  will  be  negative  when  h  is 
negative,  i.e.  when  the  curve  is  concave  downwards  (§  46,  p.  75). 
The  same  remarks  apply  to  R,  since  R  =  1/K. 


EXERCISES   XXXVin.  —CURVATURE 

1.  Calculate  the  coi'vature  K  and  the  radius  of  curvature  i?  =  \/K  for 
each  of  the  following  curves  : 

(a)     y  =  x\     Ans.    i?  =  (1  +  4  a-2)3/2/2. 

(6)     y  =  x3.     Ans.    i?  =  |  (1  +  9  .r*)3/-76  x|. 

(c)  2/  =  ax2  +  6x  +  c.     Ans.   i?  =  |[1  +  (2  ax  +  6)2]3/2/2  a\. 

(d)  t/^  =  4  ax.     ^HS.    i?  =  (y2  +  4  a2)3/2/4  a2. 

(e)  xy  =  a2.     Ans.   R  =  (3?  +  y^yi^/2  a^ 
(/)     y  =  3  62a;2  _  2  a;*. 

(^r)  2/  =  6  62a;2  _  ft^  +  x*. 

(/i)  2/  =  sin  X. 

(t)  2/  =  cos  X  —  (cos  2  x)/2. 

U)  y  =  e'. 

(^-)  2/  =  (e^  +  6-^/2  =  cosh  X. 

0-2        ,,2  /3.2        „2\3/2 

(m)   Vx+V2/  =  Va.     ^»is.    B  =  (x +  7jy^y(2Va). 
(n)  x3  +  2/3  -  «^-     ^"s-   -?^  =  sVaxy. 
(0)  2/  =  ^Ce^''"  +  e-^''«)/2.     ^?is.   i2  =  2/VI«l- 

2.  The  center  0/  curvature  §  of  a  curve,  corresponding  to  a  point  P, 

is  obtained  by  drawing  the  normal  at  P  and  laying  off  B  on  this  normal 

toward  the  concave  side  of  the  curve.     Show  that  the  coordinates  of  Q 

are 

a  =  x  —  B  s'lTKf);  p  =  y  +  B  C0B<p, 

where  0  =  tan-  ^dy/dx  =  tan-im.     Show  that  these  equations  may  also 
be  written  in  the  form  : 


172  TRANSCENDENTAL  FUNCTIONS  [VI,  §97 

0 

where  m  =  dy/dx,  b  =  d^y/dx-.     [See  also  §  155.] 

3.  Assuming  the  formulas  of  Ex.  2,  find  Q  in  terms  of  P  for  each  of 
the  curves  in  Ex.  1. 

4.  Plot  the  curve  y  =  x'^ ;  draw  several  of  its  normals,  and  lay  off  a 
distance  equal  to  B  on  each  of  them  toward  the  concave  side  of  the 
curve. 

The  locus  of  these  centers  of  curvature  is  called  the  evolute  (see  §  153). 
Draw  the  evolute.        ^ 

[Note.  The  equation  of  the  evolute  may  be  found  (§  153)  by  elimi- 
nating X  and  y  between  the  equations  for  a  and  /3  and  the  equation  of 
the  given  curve.] 

5.  Plot  the  evolute,  as  in  Ex.  4,  for  the  curve  1(e)  taking  a  =  1  ;  for 
1(h)  ;  for  1(A;) ;  for  1(1),  taking  a=b  =  l. 

6.  Find  the  values  of  K,  B,  a,  /3  for  each  of  the  following  pairs  of 
parameter  equations : 

(a)  x  =  a  cos  t,   y  =  a  sin  t.     Ans.   B  =  \a\. 

(b)  x  =  acos^t,   y  =  asm^t.    A7is.    B  =  S\a  (sm2  t)/2\. 

(c)  x  =  a(e  -sine),   y  =  a(l  —  cose).     Ans.   i?  =  4|a  sin(^/2)|. 

(d)  x^fi,  y  =  t-  tys.    Ans.   B  ^  (1  +  t'')y2. 

7.  Show  that  the  radius  of  curvature  i?  of  a  curve  whose  equation  is 
given  in  polar  cooi'dinates  is 

B  = 


[p-2+(dp/dd)^-y/i 


\p^  +  2(dp/dey  -  p((Pp/de-^)\ 

[Hint.     Since  B  =  ds/da,  and  a  =  ^  + 1/-,  §  96,  p.  167,  we  have  da  =  dd  ^ 
+  d^/  ;   since  \f/  =  tan-i  [p/(fZp/d^)],  dip/dd  can  be  found.     Again,  since 
X  =  p  cos  e,  y  =  p  sin  6,  ds^  -  dx-  +  dy^  -  dp"^  +  p-  dO-.     Combine  these  to 
find  ds/da.'\ 

8.    Assuming  the  formula  of  Ex.  7,  find  the  radius  of  curvature  of 
each  of  the  following  curves  : 

(a)  p  =  a^.  (d)  p  =  cos  e.  (g)  p  =  a(\  +  cos  6). 

(b)  p  =  e^.  (e)  p  =  sin  3  0.  (h)  p  =  2/(1  +  cos  6). 

(c)  pd  =  a.  if)  P  =  a  sec  2  6.  (i)    p  =  a  +  b  cos  6. 


VI,  §98]  LIST  OF  DIFFERENTIALS  173 

98.  Collection  of  Formulas.  For  convenience  of  reference, 
and  for  use  in  the  exercises  which  follow,  we  collect  here  the 
formulas  proved  in  this  chapter.  For  convenience  in  printing 
they  are  given  in  differential  notation.  The  formulas  in  ordinary 
type  can  all  be  obtained  feadily  from  those  in  black-faced  type. 

DIFFERENTIALS   OF  TRANSCENDENTAL  FUNCTIONS 

[VIII]  d\og,x=^.-^  =  '^-log,e,  [J/=log,oe  =  0.4343]. 

X     logio-B      X 

rv¥¥¥  n     ^  i„„  ^     (t-^  [XI]         d  COS  ie=— sin  ocdx, 

[Villa]      a  lOge  OC .  ■-       -^ 

^  [XII]        rf  tan  ic  =  sec2  x  dx. 

[IX]  dB^  =  B- log,  Bdx.  [XIII]     cZctnic  =  -csc2icrte. 
[IXa]                de^  =  e^  dx,  [XIY^    d  sec  x  =  sec  x  tan  x  dx. 

[X]  dsmx  =  coHxdx.  [XV]    fZcsc.x=— cscicctn.rdcc. 

[XVI]  d  sin-i  X  =  : ,      (sin-^ x  in  1st  or  4th  quadrant). 

[XVII]  d  cos~^  X  =  ,      (cos~^ic  in  1st  or  2d  quadrant). 

VI  —  ar 

[XVIII]  dtaii-ia;=j^^,         (all  values  of  a;). 

[XIX]  d  ctn-^  X  =  ^ — 5,  (all  values  of  x). 

[XX]  d  sec-;  X  = ^^         (sec-^  x  in  1st  or  3d  quadrant). 

—  dx 
J [XXI]     d  csc~^  x  =        ,  .^         >    (csc'^ic  in  1st  or  3d  quadrant). 

[XXII]  d  vers~^  x  =  :>    (vers~^x  in  1st  or  2d  quadrant). 

Other  rules :  See  also  algebraic  forms  (p.  40  or  p.  52),  hyper- 
bolic function  forms  (Ex.  4,  p.  140,  Ex.  8,  p.  141,  and  Exs.  11- 
13,  p.  I(i6) ;   Gudermannian  (Ex.  10,  p.  166). 

Litegral  formulas :   See  Chapter  VII,  and  Tables,  IV,  A-H. 


CHAPTER  VII 

TECHNIQUE  —  TABLES  —  SUCCESSIVE  INTEGRATION 

PART  I.     TECHNIQUE   OF   INTEGRATION 

99.   Question  of  Technique.     Collection  of  Formulas.     The 

discovery  of  indefinite  integrals  as  reversed  differentials  was 
treated  briefly,  for  certain  algebraic  functions,  in  Chapter  V. 
We  proceed  to  show  how  to  integrate  a  variety  of  functions, 
but  the  majority  are  referred  to  tables  of  integrals,  since  no 
list  can  be  exhaustive.     See  Tables,  IV,  A-H. 

To  every  differential  formula  (pp.  52,  173)  there  corre- 
sponds a  formula  of  integration : 

if  d<l>(x)  =  /(ic)  dx  then   Cfix)  due  =  <t>(oc)  +  C 

The  numbers  assigned  to  the  following  formulas  corre- 
spond to  the  number  of  the  differential  formula  from  which 
they  come.  The  most  important  ones  are  set  in  black-faced 
type,  except  that  black-faced  type  is  not  used  when  the 
formula  is  easy  to  remember  intuitively.  Certain  omitted 
numbers  correspond  to  relatively  unimportant  formulas. 

FUNDAMENTAL  INTEGRALS 

mif  ^  =  0,    then    y  =  constant.    [See  §  58,  p.  99.1 
»  doc 

[The  arbitrary  constant  C  in  each  of  the  other  rules  results  from  this  rule.] 
[II]i  jk  f(x)  dx  =  k^f(x)  dx  +  a 

[III]i        J" \f{x)  +  <l>{x)\dx  =  j*/(a;)  dx  + 1«^ {x)  dx  +  C, 
174 


VII,  §99]  FUNDAMENTAL  FORMULAS  175 

[IV] i       Ja;» dx  =  ^~  +  C,  ivhen  n^-1.     (See  VIII.) 

[VI]i  uv  =  Cd  (uv)  =  ju  dv  +  Cv  dii  +  C.    ["  Parts"] 

[The  corresponding  formula  [V]i  for  quotients  is  seldom'  used.    See  §  103.] 
[VlIJi  i'f{ii)du]       .      =.r/[^(a-)]rf^(ir)  +  C 

x/  J.l«  =-(/>(«)         «/ 

[Substitution]  =  Cf[<t>{x)  ]  ^?^  dx  +  C. 

J    *  dx 

[Vlllji      p^  =  log  ar  +  C.        [IX],    Jfe«^  dx    =e^+C, 

[X]i        j  cos  X  dx  ==  sin  «  +  C     [XI],     Jsin  xdx  =  -(i<^x+C. 

[XII]i  Jsec2  a;  dx  =  tan  a;  +  C 

[XIII]i  j*csc2a;f?x-  =  -etna;+a 

[XlVJi  Tsec  X  tan  a-  dx  =  sec  x  +  C. 

[XV]i  J  CSC  X  ctn  a;  dx  =  —  esc  x  +  C. 

[XVI]i  f-^^^=  =  sin-i  x  +  C  =  -  cos-^  x  +  O.    [XVII], 

[XVmji  f^^^  =  tan-i «  +  C  =  -  ctn-i  x  +  C.     [XIX]i 

»/   1  +  X" 

[XX]i  r ^^:^^ =  sec-i x-\-C=-  csc'^ x  +  C.      [XXIJ, 

•^  X  Vx^  —  1 

[XXIIJi      f—^^--  =  vers-i  x  +  O  =  -  covers-*  x  +  C". 
*^  V2x*— ar^ 

The  remaining  differential  formulas  referred  to  on  p.  173 
give  rise  to  other  integral  formulas  ;  these  will  be  found  in 
the  short  Table  of  Integrals,  Tables,  IV,  A-H. 


176  TECHNIQUE   OF  INTEGRATION       [VII,  §  100 

100.  Polynomials.  Other  Simple  Forms.  The  rules  [II], 
[III],  [IV]  are  evidently  sufficient  without  further  explana- 
tion to  integrate  any  polynomials  and  indeed  many  simple 
radical  expressions.  This  work  has  b&en  practiced  in  Chapter 
V  extensively. 

Attention  is  called  especially  to  the  fact  that  the  rules  [II] 
and  [III]  show  that  integratior*  of  a  sum  is  in  general  simpler 
than  integration  of  a  product  or  a  quotient.  If  it  is  possible,  a 
product  or  a  quotient  should  be  replaced  by  a  sum  unless  the 
integration  can  be  performed  easily  otherwise.  Thus  the  in- 
tegrand (1  +  af)/x  should  be  written  \/x  -\-  x\  (1  +  ^Y  should 
be  written  1  +  2  cc-  +  aj* ;  and  so  on.  This  principle  appears 
frequently  in  what  follows. 

101.  Substitution.  Use  of  [VII].  As  we  have  already  done 
in  simple  cases  in  Chapters  V  and  VI,  substitution  of  a  new 
letter  may  be  used  extensively,  based  on  Rule  [VII]. 

Example  1.     To  find    ^      ^^ 


Va-  -  x^ 
Set  u  =  x/a,  then  du  =  dx/a,  ov  dx  =  a  du,  and 

r^^=sm- 


dx        _C      adu        _.       ..       -"—'„+ (7=  sin-i--fC. 


Va2  -  x2      •^  Va-^  -  a^u^ 


Check.        d  sin-i  -  —. 


dx/a    _       dx 


V-S  ""  V' 


Va^  -  x2 


Example  2.     To  find    j  sin  2  x  dx. 
Method  1.     Direct  Sxchstitution. 

j  sin  2xdx  =  |  J  sin  (2  x)  d  (2  x)  =  -  |  [cos  2  x  -|-  C]  =  —  |  cos  2  x  -f-  C. 
Check,     d  (—  I  cos  2  a;)  =  -  ^  (?  cos  (2  x)  =  -f- 1  sin  (2  x)  d  (2  x)  =sin  2  x  dx. 
Method  2.     Trigonometric  Transformation  and  Stihstitution. 
i  sin 2x dx  =  j  2 sin x cos xdx  ——2  I  cos x d(cos x) 
=  —  (cos  x)'^  +  /r=  —  cos^x  4-  K. 


Vll,  §  102]  SUBSTITUTION  177 

Notice  that  cos- a;  +  ^  =  1/2  cos  2  a;  +  C  since  cos  2  x  =  2  cos^  x  —  \. 
Do  not  be  discouraged  if  an  answer  obtained  seems  different  from  an 
answer  given  in  some  table  or  book  ;  two  apparently  quite  different 
answers  both  may  be  correct,  as  in  this  example,  for  they  may  differ  only 
by  some  constant.* 

Whenever  a  prominent  part  of  an  integral  is  accompanied  by 
itn  derivative  as  a  coefficient  of  dx,  there  is  a  strong  indication  of 
a  desirable  substitution ;  thus  if  sin  x  occurs  prominently  and 
is  accompanied  by  cosxdx  substitute  w  =  since;  if  log  x  is 
prominent  and  is  accompanied  by  (l/x)dx,  set  u  =  \ogx;  if 
any  function  f(x)  occurs  prominently  and  is  accompanied  by 
df(x),  set  u=f{x).  This  is  further  illustrated  in  exercises 
below. 

102.  Substitutions  in  Definite  Integrals.  In  evaluating  defi- 
nite integrals,  the  new  letter  introduced  by  a  substitution  may 
either  be  replaced  by  the  original  one  after  integration,  or  the 
values  of  the  new  letter  which  correspond  to  the  given  limits 
of  integration  may  be  substituted  directly  without  returning 
to  the  original  letter. 

Example  1.     Compute  J  ^~o     sin  x  cos  x  da;. 


3Iethod  1. 

fi=n/2    .                                         rx=TT/2             -I                       y2-lx=»r/2         SlVl^  Xl'='^/^ 

\          sinxcosxc?x=  \          udu             =—            ^  ^HLjf 

J-»=0                                               Jx=0                  Ju=sinx         2j,=o                  2      Jx=0 

1 

"2' 

Method  2, 

rx=7T/2    .                       ,            ru=l            -1                     M2-lt4=l        1 

\          sm  X  cos  xdx=  \       n  du             =  ^         =  -, 

J»=0                                               J«=0              J«=8inx         2j„=o        2 

since  ?<(=  sin  x)  =  0  when  x  =  0,  and  u  =  l  when  x  =  7r/2. 

Care  must  be  exercised  to  avoid  errors  when  double-valued  functions 
occur.  The  best  precaution  is  to  sketch  a  figure  showing  the  relation  be- 
tween the  old  letter  and  the  new  one.  In  case  there  seems  to  be  any 
doubt,  it  is  safer  to  return  to  the  original  letter. 

*  Occasionally  it  is  really  difficult  to  show  that  two  answers  do  actually 
differ  by  a  constant  in  any  other  way  than  to  show  that  the  work  in  each  case 
is  correct  and  then  appeal  to  the  fundamental  theorem  (§  58,  p.  90). 


178  TECHNIQUE  OF  INTEGRATION       [VII,  §  102 

EXERCISES  XXXIX.  —  ELEMENTARY  INTEGRATION      SUBSTITUTION 

1.  Integrate  the  following  expressions  : 

(a)    ((l  +  x)(l+x^)dx.  (e)    j (e^  +  e-'^ydx. 

(c)  j*(a  +  te)2dx.  (g)    ((l  +  2x)Wxdx. 

(d)  J^^^^^^dx.  (h)    r(x2-2)(a;i/2  +  a;2/3)dx. 

2.  In  the  following  integrals,  carry  out  the  indicated  substitution ;  in 
answers,  the  arbitrary  constant  is  here  omitted  for  convenience  in  printing. 

(a)  r V2X  +  S dx;  setu  =  2x  +  3.     Ans.   u^'^/S  =(2x  +  3)3/2/3- 

(b)  C-^  =  ilog(2a:  +  3)=logV2xT3. 

(c)  C ^ =itan-i(2x-f  3). 

^  ■'    Jl+(2x  +  3)2      ^  ^  ^ 

(d)  (x  VT^dx  ;  set  M  =  1  +  x2.     Ans.    u^/ys  =  (1  +  x2)3/2/3. 

(6)       j'^^=|l0g(l+x2)=l0gVrT^^. 

(/)  fsinxVcosxdx;  set  tt=cosx.    Ans.   —2u^/^/S=-2(cosy^x)/S. 

(^)  rcosxViSxdx  =  2(sin3/2a;)/3.  (h)    ^ e^+^'x  dx  =  I  e^+-\ 

(i)  Tcos  X  (1  +  2  sin  X  +  3  sin2  x)dx  =  sin  x  +  sin^  x  +  sin^  x. 

(j)  rsin3xdr.=  CsinxCl  -  cos2x)dx  =-  cosx  +(cos3x)/3. 

(A:)  fcos  (2  X  +  3)  sin  (2  x  +  3)  dx  =  ^  sin2  (2  x  +  3) . 

(0  rsin(l-3x)cos3/2(l-3x)dx  =  T2jCos5/2(l-3x). 

(m)    f_^_;setM=--    Ans.    -  tan-i m  =  i tan-i  - . 
^       J  a-  +  x'^  a  a  a  a 


VII,  §  102]  SUBSTITUTION  179 

3.    In  the  follo^Ying  integrals,  find  a  substitution  by  inspection  and 
complete  the  integration : 

(«)     fT-^=Tlog(3a;  +  4)=log(3x  +  4)i/3. 
J  8  X  +  4      3 

(6)     CV\-2xdx  =  -(1-2  a;)3/2/3. 
(c)      rsin(2x-3)dx  =  -^cos(2x- 3). 

(e)  I  cos''  X  dx  =  sin  X  —  (sin^  x)/3  . 

(/)  Tecs  xsinSxdx  =  (sin*  a-)/4.        (g)   p^^  dz  =  j  (log  xy. 

(ft)  j'2xcos(l +x2)dx  =  sin(l +x2). 

(i)  i  tan  X  sec2  xdx  =  (tan^  x)/2. 

(j)  C(e2'  +  e-^)dx  =(e'-^dx+(e-'-^dx  =  (e^' -  e-'^) /2. 

{k)  fcos^  .r  dx  =  j*  [  (1  +  cos2  x)/2]  rfx  =  x/2  +  (sin  2  x)/4. 

(l)  fsin2xdx=  (*[(!  -  cos 2x)/2]  dx  =  x/2 -(sin2x)/4. 

(m)  jcoss  X  dx  =  sin  x  —  2  (sin^  x)/3  +  (sin^  x)/5. 

(?j)  fctnxdx  =  i  (cosx/sinx)  dx  =  log  sin  x. 

(o)  jtan  X  dx  =  —  log  cos  X  =  log  sec  X. 


dx  1 


■tan-i 


2  +  x-      Sy/2  VV2/ 

*^V'l2-4x2      2Jv'3-x''^      ^  \V3/ 


180  TECHNIQUE  OF  INTEGRATION       [VII,  §  102 

4.    Compute  the  values  of  the  following  definite  integrals  : 
«)     r'^  =  ^tan-^-^1^^  =  J-tan-iJ-  =  ^^. 

b)  r'      ''"       =sin-i(-^^1-=^^sin-i-L  =  !r. 

c)  f '^'^i^o  =  i  ^og,  (3  +  a:2)1'='  =  i  (log,4  -  log,3)  =  .1438. 

d)  r^'    ^lfl_:=-v/23T^1^'^-(Vl-V2)^.4142. 

-'xM)    V2  -  X2  Jx=0 

e)  \'~  '  sin^xdx  =  ["— cosx  +  (cos3a;)/3T"''''  =5/24. 
/)    C^\^e''  dx  =  e^ysl'"^  =  e/3  -  1/3  =  .5728. 

•^1=0  Ji=0 

>^x=i  ^        J^=o6  +  2x2 

Cx=Tl%  /•  1=77/4 

f)  i  sin^xcosxdx.  (m)    (  cos^xdx. 

rx=:-ir/6  /•x=TT/i 

j)     \  sin2xdx.  (n)     \ 

Jx=0  •^z=0 


cos^xdx. 

x=0 

5.  Find  the  area  under  the  witch  y  =  l/(a  +  bx^)  for  a  =  9,  &  =1, 
from  X  =  0  to  X  =  1 ;  for  o  =  8,  6  =  2,  from  x  =  1  to  x  =  10. 

6.  Find  the  volume  of  the  solid  of  revolution  formed  by  revolving  one 
arch  of  the  curve  y  —  sinx  about  the  x-axis. 

7.  Find  the  area  under  the  general  catenary 

y  =  a  cosh  (x/a)  =  a(e^''«4-e-*/'»)/2 
from  X  =  0  to  X  =  ffl. 

8.  Find  the  area  of  one  arch  of  the  cycloid 

x  =  a{9  —  sin^),  y  =  a{l  —  cos^). 

9.  Find  the  volume  of  the  solid  of  revolution  formed  by  revolving  one 
arch  of  the  cycloid  about  the  x-axis. 


VII,  §103]  INTEGRATION  BY  PARTS  181 

10.  Compare  the  area  of  one  arch  of  the  curve  y  —  sinx  with  that  of 
one  arch  of  the  curve  y  =  sin  2  a; ;  with  that  of  one  arch  of  j/  =  sin2  x. 

11.  Show  how  any  odd  power  of  sin.x  or  of  cos  a;  can  be  integrated  by 
the  device  used  in  Ex.  2,  (j). 

12.  Show  how  any  power  of  sin  x  multiplied  by  an  odd  power  of  cos  x 
can  be  integrated. 

103.   Integration  by  Parts.     Use  of  Rule  [VI].  —  One  of  the 

most  useful  formulas  in  the  reduction  of  an  integral  to  a  known 
form  is  [VI],  which  we  here  rewrite  in  the  form 


vdu 


[VI']  \udv  =  uv—  j 

called  the  formula  for  integration  by  parts.     Its  use  is  illus- 
trated sufficiently  by  the  following  examples  : 

Example  1.     J  x  sin  a;  dx.    Put  u  —  x,dv  =  sin  x  dx ;  then  du  =  dx  and 
r  =  J  sin  X  dx  =  —  cos  x  ;  hence, 

J  X  sin  X  dx  =  —  x  cos  x  +  J  cos  x  dx  =  —  x  cos  x  +  sin  x  ;  (check). 

Example  2.   J  log  xdx.     Put  log  x  =  u,  dx  =  dv;  then  dzi  =  (l/x)dx, 
V  =  x,  and 

flog  xdx=x  logx—  rx---dx  =  X  logx—  fdx  =xlogx— x  +C;  (check). 

^  X 

Example  3.     J  Va-—x-  dx.     Put  u  =  Va^—x'^,  dv  =  dx;  then  v=x,  and 

du  =  -^=3;  dx,  and  \  Va-^  -  x^  dx  =  x  Va^  -  x^+  \     ,  ; 

Va^  -  x2  -^  -^  Va2  -  x2 

but,  by  Algebra,       C ^^  =-  C V^^TZT^^  dx  +  f-^^^ ; 

'hence 

a^dx 


Va2  -  ic'-  dx  =  xVa^  —  x^  +  j  — = 


This  important  integral   gives,  for  example,  the  area  of   the  circle 
x2  +  2/2  —  (i2,  since  one  fourth  of  that  area  is 


182  TECHNIQUE  OF  INTEGRATION      [VII,  §  103 

EXERCISES  XL.— INTEGRATION   BY  PARTS 

1.    Carry  out  each  of  the  following  integrations : 

a)  ix  Gosxdx  =  X  sin  x  +  cos  a;  +  C. 

6)  j  xe^  dx  =  e^(a:  —  1)  +  C     [Hint,     u  =  x,  dv  =  e'  dx.] 

c)  Ix  log xdx=—  xV4  +  (x2  log x)/2  +  C. 

d)  rx2  log  xdx  =  -  xV9  +  (x3  log  x)/3  +  C. 

e)  fxV  d-K  =  e^(x2  _  2  X  +  2)  +  C.     [Hint.     Use  [VI]  twice.] 

/)  I  sin-i  X  dx  =  X  sin-i  X  +  Vl  —  x^  +  C.     [Hint.     M  =  sin-ix.] 

g)  \  tan-^  xdx  =  x  tan-i  x  —  log(l  +  x-)'^'^  +  C 

h)  Cxs  tan-i  x  dx  =  (x^  tan-i  x)/3  -  xV6  +  log(l  +  x-y^  +  C 

i)  i  x(e^  —  e-*)/2  dx  =  |  x  sinh  xdx  =  x  cosh  x  —  sinh  x  +  C. 

j)  rx2  e2x  (^a;  =  e2^(x2/2  -  x/2  +  1/4)  +  C. 

k)  fe^  sin  x  dx  =  e^(sin  x  —  cos  x)/2.     [Set  m  =  e=^ ;  use  [VI]  twice.] 

I)  Ce^""  cos  2  X  dx  =  e^{2  sin  2  x  +  3  cos  2  x)/13. 

m)  I  e-^  sin  4  X  dx  =  —  e-"^(4  cos  4  x  +  sin  4  x)/17. 

«)  I  e"^  cos  MX  dx  =  e«^(n  sin  nx  +  a  cos  nx)/(a'^  +  n'^) . 

2.  Show  that  |  P(x)  tan-i  x  dx,  where  P(x)  is  any  polynomial,  re- 
duces to  an  algebraic  integral  by  means  of  [VIJ.  Show  how  to  integrate 
the  remaining  integral. 


VII,  §103]  INTEGRATION   BY  PARTS  183 

3.  Show  that  i  P  (x)  log  x  dx,  where  P(x)  is  any  polynomial,  re- 
duces to  an  algebraic  integral  by  means  of  [VI].  Show  how  to  integrate 
the  remaining  integral. 

4.  Express  |  x"  e<"  dx  in  terms  of  |  x"-i  e<"  dx.  Hence  show  that 
I  P{x)e'"dx  can  be  integrated,  where  P{x)  is  any  polynomial. 

6.   Carry  out  each  of  the  following  integrations : 
(«)    fci  4-2x-x2)  logxcZx.  (c)    r(.>;2-2.T  +  3)e-2'dx. 

(6)    r(3a:2  +  4x-l)tan-ia:rfx.  {d)    ("(xs  -  5)e*^dx. 

6.  From  the  rule  for  the  derivative  of  a  quotient,  derive  the  formula 
(  {l/v)du  =  u/v  +  I  {u/v-)dv.     Show  that  this  rule  is  equivalent  to  [VI] 

if  u  and  v  in  [VI]  are  replaced  by  1/v  and  u,  respectively. 

7.  Integrate  i  e^  sin  x  dx  by  applying  [VI]  once  with  u  —  e^,  then 
with  M  =  sin  x,  and  adding. 

8.  Integrate  i  e""  sin  nx  dx  by  the  scheme  of  Ex,  7. 

9.  Find  the  values  of  each  of  the  following  definite  integrals : 


(a)    C^logxdx. 

(d)    r^\l +3x2)  tan-ixrfx. 

{h)   J^^'xe-dx. 

(e)    r*^^ V2^  cos  3  X  f?x. 

(c)    \         sin-i  X  dx, 

Jx=Q 

(/)   1^^%' -  e-)  dx. 

10.  Find  the  area  of  one  arch  of  the  curve  y  =  e~'  sin  x. 

11.  Find  the  area  beneath  each  of  the  following  curves  :  (a)  y  =  xe-', 
(6)  y  =  x-e-',    (c)  2/  =  x^e~^,  from  x  =  0  to  x  =  1. 

12.  Compare  the  area  beneath  the  curve  y  =  log  x  from  x  =  1  to  x  =  e 
with  the  approximate  result  obtained  by  using  the  adaptation  of  the 
prismoid  formula,  Ex.  6,  p.  128. 

13.  Show  that  the  sum  of  the  area  beneath  the  curve  y  =  smx  from 
X  =  0  to  X  =  A;  and  that  beneath  tlie  curve  y  =  sin-i  x  from  x  =  0  to  x  = 
sin  k  is  the  area  of  a  rectangle  whose  diagonal  joins  (0,  0)  and  (k,  sin  k). 


184  TECHNIQUE  OF  INTEGRATION       [VII,  §  104 

104.  Rational  Functions.  To  integrate  any  polynomial, 
rules  [II],  [III])  [IV]  are  sufi&cient.  When  the  integrand  is 
rational,  but  fractional,  the  formulas  [VIII]  and  [XVIII]  are 
required. 

Example  1. 


/(^ 


2x 1 \dx  =  x^  —  log  x-\-2,  arc  tana;  +  C. 

X      1  +  x^J 


Moreover,  it  may  be  necessary  to  prepare  the  expression  for 
integration. 

Example^.     ^^2.^  +  ar-  +  2.-1^3^_l  2 

^  D  Q^  +  X  X       l+x" 

The  method  of  preparation  is  as  follows  :  Any  rational  frac- 
tion N/D  in  which  the  degree  of  N  is  greater  than  or  equal  to 
the  degree  of  D  may  be  replaced  by  an  integral  quotient  Q  to- 
gether with  a  proper  fraction  R/D,  where  R  is  of  lower  degree 
than  D.  This  results  from  ordinary  algebraic  division,  where 
Q  is  the  quotient  and  R  is  the  remainder.     Thus, 

N^2x*-\-x^  +  2x-l^^^     x^-2x-\-l 
D  x^-\-x  x"  +  X 

The  rational  fraction  R/D  can  then  be  broken  up,  by  algebra, 
into  a  sum  of  proper  partial  fractions,  whose  denominators  are 
the  real  factors  of  the  first  or  second  degrees  of  D,  and  integral 
powers  of  these  factors.  If  D  has  only  simple  factors,  there  are 
just  as  many  proper  partial  fractions  as  there  are  factors,  each 
partial  fraction  taking  as  its  denominator  one  factor. 

Thus  in  Example  2,  the  factors  of  D  are  x  and  x^  +  1;  hence  we  write 
B  _x^-2x  +  1^A  J  Bx+  O 

D  x^  +  x  X       \+x^ 

where  A^  B,  C  are  at  present  unknown  constants.* 

*  Notice  that  the  uumerators  inserted  are  just  one  less  in  degree  than  the 
correspouding  denominators :  this  is  because  it  is  known  that  the  resulting 
partial  fractious  will  be  proper  fractions.  The  numerator  should  be  written 
in  the  most  geueral  form  for  which  the  fraction  is  a  proper  fraction. 


VII,  §  104]  RATIONAL  FUNCTIONS  185 

Clearing  of  fractions,  we  find 

comparing  the  coefficients  of  like  terms, 

A  +  B=l,  C=:-2,  A  =  l,  whence  ^  =  1,B  =  0,  C=-2 

aud  -  -  =  ^"-2^  +  1  =1  +  _JIl1_ ,     (Check  by  addition) 

D  a^  +  x  X      1  +  a;2      ^  ^ 

whence  the  example  can  be  completed  as  above. 

If  D  consists  of  one  factor  raised  to  an  integral  power,  and 
if  the  degree  of  R  is  not  less  than  the  degree  of  that  factor, 
then  B/D  can  be  simplified  still  further  by  ordinary  division, 
using  the  factor  to  the  first  power  as  a  divisor. 

Example  3.    Given  -  =  ^x  +  2     ^^  ^^^  C  R  ^^ 
D      (x+l)2  JD 

Dividing  i?  by  (x  +  1),  we  find 

x+l~         x  +  1  x  +  l' 


hence  ^  =  -^^  ■  — !—  =  -^ ;  (check). 

D      x  +  l      x  +  l      x  +  l      (x  +  l)2'  ^  ^ 

Therefore 

r|dx=r-^-r-^^  =  31og(x  +  l)+^  +  C;  {check). 
J  D  J  X  +  1      J  (x  +  1)-^  ^  ^     X  +  1 

Example  4.     Given  ^  ^3:^  +  2x2  +  2    ^^  ^^^  C^ ^^^ 
D         (x2  +  1)2    '  J  D 

i 

Dividing  B  by  (x^  +  1),  we  find 

R 


=  x  + 


X2+1         "     ■  x2+r      D         (X2+1)         (x2  +  l)2' 

and       (^  dx  -  f -^-^  +  {^^ -  r     ""^^ 

JD  Jx2  +  l^J.r2+l        J(X2+1)2 

=  I  log  (x2  +  1)  +  2  tan-1  x  +     J        ;  {check). 


186  TECHNIQUE  OF  INTEGRATION       [VII,  §  104 

If,  finally,  D  has  several  factors,  some  repeated  and  some 
simple,  we  proceed  as  before  to  make  as  many  distinct  proper 
partial  fractions  as  there  are  distinct  factors,  using  as  denomi- 
nators the  simple  factors  as  before,  and  the  repeated  factors 
raised  to  the  same  poiver  as  that  in  zvhich  they  occur  in  D.* 

Example  5.     Given   ^  =  i^i±l^i±l^i  +  l^±^ ,  to  find    (^dx. 
D  (3:k  +  2)(x2  +  1)2  Jd 

Set  —  =       a        ,  'bv^  +  cofi  +  dx  +  e 

D      3x  +  2  (x2  +  l)2 

clear  of  fractions,  and  equate  the  coefficients  of  like  powers  : 

a  +  36  =  4,  26  +  3c  =  8,  2«  +  2c  +  3cZ=6,  2d  +  3e  =  6,  a  +  2e  =  5; 

whence  a  =  1,  6  =  1,  c  =  2,  d  =  0,  e  =  2  j 

and  B^^^}_      x^  +  2x^  +  2^ 

D      3  X  +  2  (x2  +  1)2 

The  integration  is  then  completed  as  in  Ex.  4. 


EXERCISES  XLI.  — RATIONAL  FRACTIONS 

Carry  out  the  following  integrations  : 


^Jx2-l      2JVx-l      x  +  iy  2     "x+l 

c)  r_^^  1    log^ZlJ?. 
J  X-  —  a-     2  a        x  +  a 

d)  r ^^ =r ^ =  ltan-i^+i. 

^   J  x2  +  2  X  +  5      J  (a;  +  1)2  +  4      2  2 

')     f  .,  ,  f    ,      =  tan-i(x  +  2). 

J  X-  +  4  X  +  5 


*  This  simple  rule,  together  with  the  algebraic  reduction  by  long  division 
just  mentioned,  is  perfectly  general  and  is  always  successful  whenever  the 
denominator  D  can  be  factored. 


VII,  §1041  RATIONAL  FUNCTIONS  187 

^       Ja:-'2  +  2x-3     J  (a:  +  1)2  _  4     4     "x  +  s' 

C/i)    r  (?x  _  1  ^p„2x-2  _  1  ^Q    X-  1 

J4x-^  +  4x-8~12     °2x  +  4~12     ^x  +  2' 

2.    In  the  following  integrals,  first  prepare  the  integrand  for  integra- 
tion as  in  §  104  ;  then  complete  the  integrations. 

(a)    r_^^^ll_cZx  =  log(x  -  2)  +  2  log(x-3)  =  log  [(x-2)(x-3)2]. 
^  X-  —  5  X  +  b 

(P)     f -"^  ""  V'V^  =  3  logx  -  |log  (3  X  +  6). 
-^  0  X  +  0  X''  6 

(c)  i  •    "^     fZx  =  X  +  tan-1  x. 
^  X-  +  1 

(d)  f   ,  ^~^ dx  =  log  Vx2  +  2  X  +  2  -  2  tan-i(l  +  x). 

J  X-  +  2  X  +  2 

(e)  r ^:^ =J__tan-i-^-ltan-i*. 

^  ^     J  X*  +  7  x-^  +  12       V3  V3     2  2 

'     ^     Jx3  +  1 


^  ^  Jx2  +  10x  +  25 
(i)     r    8x8-36x2-2 


(2x  +  l)(2x-5) 
x2  -  20 


dx. 


1)  (a;2  _  4) 


x2-45 
2  x3  -  18  X 


dx. 


J  x3  +  x^--  2  X 

^  X2  +  X  +  1 


2x  + 


r2x2-x  +  3^,. 

J    X  +  X2  +  X3 

.  r        dx 

^^    Jx3  +  x2  +  X+l' 
r        T^-dx         . 

^^   J:(.-*  +  x2-2 

^     JX2-X 


J   (X  +  1)" 


'^L- 


(fx. 


3x 

=Ca:+l;2 


dx. 


188                TECHNIQUE  OF  INTEGRATION       [VII,  §  104 
3.    Derive  the  following  formulas  : 
(a)  j ^- r  =  — — ^  log- 


(rtx  +  b)  {mx  +  n)      an  —  hm        ax  +  h 


(^'fe 


X  dx  _      1      J      (x  +  aY  _ 

a)(x  +  6)  ~  a-b     "^  [x  +  by' 

xdx  _      b      1 V'x^  +  a  .     Va     t„Q-i    « 


(c)     f ^i?^!^ _=_A_log 


X  +  6        a  +  62  ^- 

4.  Derive  each  of  the  formulas  Nos.  18-24,  Tables,  IV,  A. 

5.  Evaluate  each  of  the  following  definite  integrals  : 

p28-3x-x2^^^  p-i5_^^+3x2_3^3 

^    ^     Jx=C/5        (5  X  -  2)3 

[Note.     Further  practice  in  definite  integration  may  be  had  by  insert- 
ing various  limits  in  the  previous  exercises.] 

6.  Carry  out  each  of  the  following  integrations  after  reducing  them  to 
algebraic  form  by  a  proper  substitution  : 


r     sinx      ^^  (r^^^dx.          (c)    f-^ 

^   ^  J  1  +  cos2  X  ^  ^  J  4  -  sin2  x                ^  W  1  -  e' 

(d)  fsecx^x^r  ""^^     dx  =  llogl±^''^. 
^  ^  J                   J  I-  sin2  X  2     ^  1  -  sin  X 

(e)  jcsc  X  dx.  (g)    i  csch  x  dx.            (j)    i 

f  ,..    fsinxcosx   ,^       .,.-.     re-'  — e-^ 

(/)   Jsech  X  dx.  (0  J  i^,,,3^^  ^^-      (^)  J  ,-7^:7="^ 


^^^^^-^ —  dx. 

tan  X  —  tan2  x 


dx. 


105.  Rationalization  of  Linear  Radicals.  If  the  integrand 
is  rational  except  for  a  radical  of  the  form  -x^ax  -f  b,  the  sub- 
stitution of  a  new  letter  for  the  radical, 


r  —  Vax  +  b, 
renders  the  new  intesrrand  rational. 


VII,  §106]  QUADRATIC   RADICALS  189 

Example  1.     Find   (■''  +  ^^  +  ^  dx. 

Setting  r  =  Vx  +  2,  we  have  x  =  r-  -2  and  dx  =  2rdr;   hence 


J      1  +  a;  J     ,.-  —  1  J  r  +  1 


?-dr 


=  2J'fr  +  1  ---^Vzr  =  j-2  +  2r-21og(r+  1)-|-  G 

=  X  +  2+  2  Vx  +  2  -21og  (  Va-.  +  2  +  1)+  C. 

The  same  plan  —  substitution  of  a  neiv  letter  for  the  essential 
radical — is  successful  in  a  large  number  of  cases,  including 
all  those  in  which  the  radical  is  of  one  of  the  forms : 


a-^/",  (ax  +  by/'^,  /^^^f^J+l^Y'; 


where  7i  is  an  integer.    Integral  powers  of  the  essential  radical 
may  also  occur  in  the  integrand. 

106.  Quadratic  Irrationals:  Va+bx±  x-.  If  the  integral 
involves  a  quadratic  irrational,  either  of  several  methods 
may  be  successful,  and  at  least  one  of  the  following  always 
succeeds : 

(A)  If  the  quadratic  Q  =  a-\-bx±x-  can  be  factored  into 
real  factors,  we  have 


±x_ 


VQ  =  ^{a  +  x){^3±x)  =  (a  +  x)^J^^^ 

^  a  +  x 

and  the  method  of  §  105  can  be  used.  The  resulting  expres- 
sions are  sometimes  not  so  simple,  however,  as  those  found  by 
one  of  the  following  processes. 

(B)  If  the  term  in  x^  is  positive,  either  of  the  substitutions 

VQ  =  t-\-x,  ^Q  =  t-x, 

will  be  found  advantageous.  One  of  these  substitutions  may 
lead  to  simpler  forms  than  the  other  in  a  given  example. 


190  TECHNIQUE  OF  INTEGRATION       [VII,  §  106 

(C)  Completing  the  square  under  the  radical  sign  throws 
the  radical  in  the  form 

VQ  =  V±k±{x±  cY; 

the  substitution  x  ±c=y  certainly  simplifies  the  integral,  and 
may  throw  it  in  a  form  which  can  be  recognized  instantly. 

Example  1.     Let  VQ  =  Va^  ±  a^ ;  show  the  effect  of  substi- 
tuting ^Q  =  t  —  x. 


If  Va?±a-  =  t  —  x,  we  find 


^=^'-='-^'*'^— ^=^^ 


and  the  transformed  integrand  is  surely  rational.  Carrying 
out  these  transformations  in  the  simple  examples  which  follow, 
we  find 


(i) 


J  V^c^±^     J\  2f  2t    J         J   t         ^    ^ 

=  log  (£c  +  VQ)  +  C,         where  Q  =  x?±a^. 


ai)  fv^..^J'lffJ:±f.u=fa.£^f^y. 


^^'±|log«  =  ^±fl»«(xWO)H-C. 


These  integrals  are  important  and  are  repeated  in  the  Table 
of  Integrals,  Tables,  IV,  C,  33,  45  a.  Many  other  integrals 
can  be  reduced  to  these  two  or  to  that  of  Ex.  3,  p.  181,  or  to 
Rules  [XVI]  or  [XX]  by  process  (C)  above. 


VII,  §106]  QUADRATIC   RADICALS  191 

EXERCISES  XLU.— INTEGRALS   INVOLVING   RADICALS 
1.    Verify  the  following  integrations : 
(a)   ^xVTT^x  dx  =^^=^  (1  +  x)3/2. 
V'2  -  x  -  1 


^'^'^7=  =  log-    

)  \/2  —  X  \/2  -  X  +  1 


2  tan-Vx-2. 

(x-1)-  ■ 


W    f 

-^  (x-1 

(0    f ^i-= 

-^  (x-DVx-i 

(,)    C ^^  ^-l,tan-iA/«^±I. 

•^  (ax  +  2  6)\/ax+ 6     aVft  ^ 

(/)    r       dx        ^      Vx  +  1  I  1  jQg  Vx  +  1  +  1 

(g)    f  («  +  to)3/2(?x  =  A  (a  +  6x)5/2. 
^  5  & 

2.   Carry  out  the  following  integrations  : 

(«)    f-^-  (&)    f^l^-  (c)    f 

-'  x-Vx  -  1  •'  Va  -X  '' 

•^  Vx  +  1  -^  Vl  -  X  -^  (x  +  2)  v^^T 

(^)i'S^3--  Wj*^.^^-  «     1(3^3- 

(,•)  1^^  .X.        (.)  J^^^x.         CO  J^  ^x. 
^^•J^'x  +  Vx  ^'     -^^^^^  +  2  ''J^x  +  1 


V  ax  +  6 


192 


TECHNIQUE  OF  INTEGRATION       [VII,  §  106 


3.    Carry  out  the  following  iutegrations  by  fii-st  making  an  appropri- 
ate substitution  : 


(a)    f-^ 
J  1  +  e=" 

(0   (^ 


dx. 


2Vcosa; 


^xdx 


3  tan  X 


id)     1^^ 

(e)     r+'^cosxdx. 
'^  1  —  Vsin  x 

,^x     r        sin  X  dx 

^'  '  J  (2-3cosx)3/2' 


,  4.    Substitution  of  a  new  letter  for  the  essential  radical  is  immediately 
successful  in  the  following  integrals  : 


(a)    i  xVl  +  x'-^dx 
a-s  d: 


^  \/2+2x+x2 

•^  Vl  +  x2  ^  ^   J  (a  +  5x-^)3/2 

(0    |x(,+x=)-.x.         (/)  St^^^^, 

5.   Cany  out  the  following  integrations  : 

(«)    r-4^  =  log(x+V^;2Zl). 


Va  +  hx^ 
dx 


hx^ 


(6)  r    ^^'^    =iiog^-^"+^'- 

(c)  r — ^£^,^=i^iz«. 

(d)  r ^^£^ 

•^  (1+2X2)V1  +X2 
-^    Vl  -  X2 


Vx2  +  1 

dx  =  sin-1  X  —  Vl  —  x2. 


(/)  j'.^^L^^^^V2tan-i- 
^  V4x^-  I  *^x\/a2_x2 


•V2 


<'•'  J^ 


dx. 


W  _r ^^^^^-       (,;)   f '^-==.      (I)    (^^- 

-^(x-l)Vl-x2  *^  (l  +  x)Vl  +  x-^  •^Vx^  +  4 


VII,  §  106]  QUADRATIC   RADICALS  193 

6.  The  following  integrations  may  be  performed  by  the  methods  of 
§  106;  note  especially  method  (C),  which  consists  in  completing  the 
square  under  the  radical. 

(a)    r  '^^  =  Iog(l  +  2  .r  +  2  Vx-  +  x  +  1  )• 

•^  Vx-  +  X  +  1 

(^,)    r  ^^  ^sin-i^^^. 

*^  Vl  +x  — X-  V5 

(c)     f        ^^         =  sin-i  ^— ^  =  vers-i  x  [  +  const.]. 
•^  V2  (Tx  —  x2  a 


•^  ./-Vl +  X  +  X-  ^ 

(.)    r  ^^  ^Isin-i^::^. 

•^xV3x2  +  4x-4      2  2x 

-^  V2x--^  +  x+l  -^  Vl+x-2x'-2  -^  \/l-2 


(?X 

+  1  -^  Vl+x-2x'-2  -^  Vl- 

(0     r  ^^'^      —  •  (0      f  Vl  +  X  +  x^  dx' 

•^  VO  X  -  x"^  -  5  •' 

(;)    r ^-^  (m)    r  V;3  X-'  +  10  x  +  9  dx. 

•^xVx-^  +  2x  +  3  *^ 

^'^   i'(x  +  4)V^T3¥^4*  ('^)    I'^v/BT^^^dx. 

7.    Integrate  by  "  Parts,"  [VI],  the  following  integrals  : 
(a)     i  X  sin-i  x  dx.  (d)    i  (3  x  —  2)sin-i  x  dx. 

(l.)    J^-^^Zx.  (.)    J'i^^^os-ixrfx. 

(c)     r.r  cos-ixdx.  (/)    r(sin-ix  +  2xcos-ix)dx. 


194  TECHNIQUE  OF  INTEGRATION       [VII,  §  106 

8.  Show  that  \  P(x)  s'm-^xdx  reduces  by  means  of  [VI]  to  an  in- 
tegral whose  integrand  contains  no  other  radical  than  Vl  —  a;^,  if  P{x) 
is  any  polynomial. 

9.  Show  that  by  means  of  the  substitution  a;  =  sin  ^  the  integrals 
(dx/Vl-x^  and   (dd  are  equivalent. 

10.   Reduce  the  following  integrals  to  trigonometric  integrals ;  then 
complete  the  integration  : 


i)    r(l  +  a^)(?3g  =(*(!+  sin  6)  dO,  if  a;  =  sin  0 


Ans.    6  —  cos  6  =  sin-i  x  —  v  1  —  x^. 


(J))     C da:  ^  f-^,  if  x  =  sing.     Ans.    tan  6  = 


(c)     (xVx:^  -  1  dx  =  Itan^  6  sec^  6  dd,  if  x  =  sec  0. 

Ans.    (tan3  0)/3  =  (x^  -  l)yyS. 


•^  Vl  +  a;2      -^ 


sec  0  tan  ^  d^,  if  a;  =  tau  0.     Aiis.   sec  tf  =  Vl  +  x^. 


11.    Reduce  the  following  integrals  to  algebraic  integrals ;  then  com- 
plete the  integration  : 

fa)    ( — ^ — =f ^^'  ,iix  =  sin  0.     [See  5  (/i).] 

J  1 -sing      -^  (l-x)Vl-x-'' 

(6)    Csec  edd=  (      ^•''      ,  if  X  =  sec  0.     [See  5(a).] 
•^  "^  Vx-  —  1 

(•     secgdg     ^  r dx ,  if  a:  =  tang.     [See  5(d).] 

^^JH-2tan2g      J  (i  +  2x-^)VrT^^ 


«  JttI^-  <"  SVT^.  <-^'  J"fT 


sec  g  d0 


VII,  §  108]  TABLES  OF  INTEGRALS  195 

107.  Elliptic  and  Other  Integrals.  If  the  essential  radical 
in  the  integrand  is  the  square  root  of  a  cubic  or  of  a  polynomial 
of  higher  degree,  or  a  cube  root  or  higher  root,  the  integrals 
are  usually  beyond  the  scope  of  this  book. 

If  the  only  irrationality  is  VQ,  where  Q  is  a  polynomial  of 
the  third  or  fourth  degree,  the  integral  is  called  an  elliptic  in- 
tegral. While  no  treatment  of  these  integrals  is  given  here, 
they  are  treated  briefly  in  tables  of  integrals,  and  their  values 
have  been  computed  in  the  form  of  tables.*    See  Tables,  V,  D,  E. 

108.  Binomial  Differentials.  Among  the  forms  which  are 
shown  in  tables  of  integrals  to  be  reducible  to  simpler  ones  are 
the  so-called  binomial  differentials : 

I  (ax"  -f  6)  "a;"'  dx. 

It  is  shown  by  integration  by  parts  that  such  forms  can  be 
replaced  by  any  one  of  the  following  combinations,  where  u 
stands  for  (okc"  +  6) : 

(1)  r  Wx""  dx  =  (^i)  mPx'"+i        -f  (Si)  r  jfP-'iC"  dx, 

(2)  j*  u-^x^dx  =  (A)  ?^p+ia;'"+^      +  (Bs)  j*  u^+^x'"  dx, 

(3)  f  mpx"*  dx  =  (As)  t<p+'a;"*+^      -f  (Bj)  j*  mPx'"+"  dx, 

(4)  r  ti^'x" dx  =  (A^)  «p+\x"'-"+i  +  (Bi)  r  u^x""-"  dx, 

where  Ai,  A^,  A^,  A^,  B^,  B^,  B^,  B^  are  certain  constants. 
These  rules  may  be  used  either  by  direct  substitution  from 

*  Some  idea  of  these  quantities  may  be  obtained  by  imagining  some 
person  ignorant  of  logarithms.  Then  J  {\/x)dx  would  be  beyond  his  powers, 
and  we  should  tell  him  "values  of  the  integral  ^  {l/x)dx  can  be  found  tabu- 
lated," which  is  precisely  what  is  done  in  tables  of  Napierian  logarithms.  Of 
course  as  little  as  possible  is  tabulated;  other  allied  forms  are  reduced  to 
those  tabulated  by  means  of  special  formulas,  given  in  the  tables.  Tables  of 
the  values  of  integrals  are  often  computed  even  though  the  integral  can  be 
found  in  terms  of  known  functions :  thus  tables  of  values  of  log  [x  +  Vx^-I- 1] 
=  !idx/y/x^+  12  are  to  be  found  under  the  name  iiirerse  hyperbolic  sine  of  x 
(=  sinh-i  x)  ;  see  p.  140,  and  Tables,  V,  C  ;  and  also  II,  H ;  IV,  C,  33. 


196  TECHNIQUE  OF  INTEGRATION       [VII,  §  10s 

a  table  of  integrals  in  which  the  values  of  the  constants  are 
given  in  general  f  (see  Tables,  IV,  I),  51-54),  or  we  may  denote 
the  unknown  constants  by  letters  and  find  their  values  by 
differentiating  both  sides  and  comparing  coeiheients. 

Example  \.     { — =A + -B  f — ,  by  (2) 

Differentiating  and  comparing  coefficients  of  x^  and  x°,  we  find  5  =  0 
and  A  —  1/6  ;  hence 

f ^^—  =  —    ^  ;  {check.) 

J  (a,;-2  +  ft)3/2      ^,V(ax-^+6) 

f       ^-'^^^ = ^ +  B  r *^ ,  by  (4). 

J(ax2  +  6)V2      («x2+6)V2  J(ax2  +  6)3/2'    -^  ^ 


Example 


and 


Here  ^  =  1/a,  5  =  -  2  6/a, 

3dx  ax2  +  2  6 


r x^ 

J  (ax2 - 


^y'^'     a2  V(ax2  +  6) 

109.  General  Remarks.  The  student  will  see  that  integration 
is  largely  a  trial  process,  the  success  of  which  is  dependent  upon 
a  ready  knowledge  of  algebraic  and  trigonometric  transforma- 
tions. Skill  will  come  only  from  constant  practice.  A  very  con- 
siderable help  in  this  practice  is  a  table  of  integrals  (see  Tables, 
IV,  A-H.  The  student  should  apply  his  intelligence  in  the  use 
of  such  tables,  testing  the  results  there  given,  endeavoring  to 
see  how  they  are  obtained,  studying  the  classification  of  the 
table ;  in  brief,  mastering  the  table,  not  becoming  a  slave  to  it. 

In  the  list  which  follows,  many  examples  can  be  done  by  the 
processes  mentioned  above.  The  exercises  which  are  starred  (*) 
may  be  reserved  for  practice  in  using  a  table  of  integrals. 

t  Such  formulas  are  called  reduction  formulas ;  many  other  such  for- 
mulas —  notahly  for  trigonometric  functions  —  are  given  in  tables  of  inte- 
grals. (See  Table  IV,  E«,  57,  60,  64,  etc.)  It  is  strongly  advi.sed  that  no 
effort  be  made  to  memorize  any  of  these  forms,  —not  even  the  skeleton  forms 
given  above.  A  far  more  profitable  effort  is  to  grasp  the  essential  notion  of 
the  types  of  changes  which  can  be  made  in  these  and  other  integrals,  so  that 
good  judgment  is  formed  concerning  the  possibility  of  integrating  given  ex- 
pressions. Then  the  actual  integration  is  usually  performed  by  means  of  a 
ta])le.  See  also  Tables,  IV,  Ea,  78,  82  (6) ;  Es,  85,  86;  Ec,  92-94;  Ed,  98,  106; 
B,  17(6),  25;   etc. 


VII,  §  109] 


GENERAL   EXERCISES 


197 


EXERCISES  XLUI.  —  GENERAL  INTEGRATION 

[As  stated  above,  the  exercises  which  are  starred  (*)  may 
be  integrated  by  use  of  tables  of  integrals.] 


-  <«>  S"ii^ 


dx. 


ips  +  l 


dx. 


dx. 


(a; -2)2  Jx*-3xa  +  3x2-x 

.,.     fSx^-  nx  +  21  ^^  ..     r    7a:2  +  7x-176 

p3_2x2  +  7x  +  4^_  r       0:^-3x4-3        ^^ 

^^J  (x2-l)2  ^•'Mx3-4x2-7x  +  10 


2.  (a)    r^^  +  \dx.  (e)     r_6^i^^. 

r27xMx.  r^-d^. 

^  ^    J  2  +  3x2  ^"^    J2  +  5x* 

^^^   Js-IT^-  ^'^    Jx(3  +  5xe)- 

3.  (a)    rxVT+^t^x.  (c)     T-^dx. 

(.)  (■^=^.<'-  «o  I— :M=. 

J  V3  X  +  o  -'  xVx  -  a 

^      4.    (a)    rx\/^^r4dx.  (c)     )-„,     ^^,       • 
^  ^   J  -^  V(a  +  6x)p 

(6)    rx\/a  +  ?>rdx.  (d)     f-^^^- 
*^  ♦^3\/x  +  2 


;va  +  bxdx. 

J  oX 

(h)   Tl  +  Vx  -  V^ 


3f7x 


5  —  7  X  +  2  x2 
x2dx 


•^^  J5  +  2X  +  X2 

^      J  X2  +  4  X  4-  2 


x2dx 
+  2x 


/)  j*x2V5 


5  +  2  X  dx. 


)    rxv'3x  +  7dx. 


V3-X 


dx. 


1+Vx 


(0     f   ,  -J'\^ 

•^  v'l  +  X  +  VI  +  , 
•^  Vl  +X  + Vl+; 


198  TECHNIQUE  OF  INTEGRATION      [VII,  §  109 

J    -i/n   J-  h'r  J 


(g)     ra;3(a  +  x2)i/3 
5.*    (a)    r — ^^ 

^    ^     J   (X2  +  5)3 

6.*    (a)   y 

J  Vx^  —  a 

f  \     C       x^dx 
^""^    J  (-7 +4x3 


dx 

x3  v'x^  —  4 

x^cZx 


7.* 


(7+4x3)2/3 

(a)    jsin*  X  dx. 

"^  ^   J2  +  sin^ 
iff)    fcos^  a  da. 


Va  +  bx 
n)    CA±^dx.  (p)   (xy/a  +  bx:^dx.      ! 

(r)      rx5(l +x3)i/3dx. 

^    J  (x2  +  3)3  ^  ^    J(x2  +  2x+5) 

r    5x-3     ^^  r_i2x+l)_<^ 

^  J  (2x2-1)-^  J  (x2+6x  + 


10)3 


dx 


'^1 


tan3  X  dx. 


_de^ 

2  -  3  cos  « 


(c)    j  sin2  X  cos*  X  dx. 


5sin^ 


{h)    rctn2  3xdx.  (i)     fsinS/s^cos^^ 

^   ^    Jo  (x-2)2                     Jo2x2-3  Jo     V4-x2 

^^^    3i  (2x-l)3''''-              •^2^V2^+^  -^1  VxM^ 

^''^   Jo  2x2  +  3                        ^2  Vi^^ITs  Ji  Vx2  -  1 

^•'''   Jo  (2  X  +  l)(x2  +  2)                       J-2  v/x2  -  8x 


dft 


^ 


VII,  §109]  GENERAL  EXERCISES  199 

9.*  Find  the  values  of  the  followiug  definite  integrals  by  using  the 
tabulated  numerical  values :  Tables,  V,  A-H  : 

Xx=1.5  rx=\A  /•i=4.1        1 

(.9)    T'"" ^— ^t_^  dx  =  p"^'  cosh  X dx.         (A)    p~"sinh.rdx. 

(i)    ('^'      ^•''       ==  cosh-i  x1"^".      (j)    r^^      <^-^       =sinh-ix1^\ 

XT=3.6         /7r  /J.-]  1=3.6 

/^^-       =  cosh-1  ? 

/•i=14.4  /7r  7.-|ar=M.4 

(0    i  ^-^        =  sinh-1  ^ 

Jx=fl      Vx2  +  9  3j^=o 

(m)    r^^°  <^g  i^(i^  30°),  Tables,  V,  D. 

Je=fl°     vi  -  (V-1)  sin2  tf 

(?i)    r^*'°  Vl-(l/4)sin2^f7^  =  £r  (^,  45°),  Tafe/es,  V,  E. 

r^^'  ^^  _.  (^,)    f-^°°Vl-.25sin^^dg. 

^^15°   Vl  -  .04  sin-^  d  •^«=^° 

^     X=o      Vr^^  Vl  -  .25  x2      *^«=*'     VI  -  .25  sin2  6 


=^ ,  if  X  =  SU1 


(r)    f"  ^  .11  -  .SQx^dx  =  r^^°  Vl  -  .36siu2^d^,  if  x  =  sin  6. 

^    '     Jx=l/2\      i_3.2  J9=30» 

^x=l/2  Vl  -  X-^  Vl  -  .49  X2  •^x=V'272     '      1  -  X^ 

[Note.     Many  of  the  exercises  in  Lists  XXXIX-XLII  may  be  used 
for  additional  practice  in  use  of  the  tables.] 


200  TECHNIQUE  OF  INTEGRATION       [VII,  §  109 

10.«  Show  that  the  even  powers  of  sin  x  can  be  integrated  by  reduction 
to  the  integral   |  sin^  x  dx. 

11.  Show  that  odd  powers  of  cos  x  can  be  integrated  readily  without  a 
table. 

12.*  Show  that  any  power  of  tan  x  can  be  integrated  by  reduction  to 
I  tan  X  (Zx  or  to   i  tan^  x  dx. 

13.  Show  that  any  even  power  of  sec  x  can  be  integrated  by  splitting  off 
one  factor  sec^  x  and  then  using  the  relation  sec^  x  =  1  +  tan^  x. 

14.  Show  that  \x'^e''dx  can  be  integrated  by  the  repeated  use  of 
Rule  [VI]. 

Hence  show  that  j  P(x)e^  dx  can  be  integrated,  if  P  (x)  is  any  poly- 
nomial. 

15.  If  \  f{x)dx  =  (p(x)  show  that  \f(x)  tan-ixdx  can  be  reduced 
to  ^[0  (x)/(l  +  x'^)]  dx  by  Rule  [VI]. 

State  a  similar  result  for  the  integral  |  /  (x)  sin-^x  dx. 

16.  Show  that  the  integrals  which  result  from  breaking  up  a  rational 
fraction  whose  denominator  has  only  simple  linear  factors  can  be  expressed 
in  terms  of  simple  powers  and  logarithms. 

17.  Show  that  a  ■  simple  quadratic  factor  in  the  denominator  of  a 
rational  fraction  gives  rise  to  a  term  in  the  final  answer  which  contains 
an  arc  tangent  or  a  logarithm. 

18.  Show  how  to  integrate  terms  of  each  of  the  following  types,  and 
show  that  no  others  arise  in  integrating  rational  fractions  :  ■' 

^  ^     J  ax  +  b  ^  ^     J  x'^  +  a'  ^•'^     J  ar^  +  hx  +  c 

(6)      r       ^^        .  (.)*    C  ^^  +  S  dx.     (hr    r      -^^  +  ^      dx. 

(c)*    r       ^'        •         (n*C  ^^  +  B  dx.     CO*    r      ^^  +  -^       dx. 


VII,  §110] 


IMPROPER   INTEGRALS 


201 


PART   ir.     T^IPROFER   AXD   MULTIPLE   INTEGRALS 

110.  Limits  Infinite.  Horizontal  Asymptote.  If  a  curve 
approaches  the  a^axis  as  an  asymptote,  it  is  conceivable  that 
the  total  area  between  the  x-axis,  the  curve,  and  a  left-hand 
vertical  boundary  may  exist;  by  this  total  area  we  mean  the 
limit  of  the  area  from  the  left-hand  boundary  out  to  any  vertical 
line  X  =  m,  as  m  becomes  infinite. 


Example  1.  The  area  under  the 
curve  y=e^''  from  the  y-axis  to  the 
ordinate  x  —  m  is 


J  1=0         t/i=0 


■  dx  =  1  —  6" 


As  m  becomes  infinite  e""*  approaches 
zero;  hence 


i            i 

.                   2/=^- 

Y    qi 

^5 

'^. 

//kS-4y  ^ 

U            J       X=--in 

A\       =   j        e~' dx=  lim  j 

-1  x=0  t/x=0  „,=x,/^=0 

and  we  say  that  the  total  area  under  the  curve  y  =  e 
a;  =  0  to  a;  =  +  oc  is  1. 


Fig.  44 
e-'(?a;=  lim  (1  —  e"'")  =  1, 

from 


Example  2.    The    area  under  the  hyperbola  y  =  1/x  from 
a;  =  1  to  a;  =  m  is 

a\  ^^  =  f"^"'^  =  log  x\  ^"  =  log  m. 

Jx=l  c/x=l         a;  Jar=l 

As  m  becomes  infinite,  log  m  becomes  infinite,  and 

I  -|  x=m  1 

lim  1-1  =  lim  log  in 


does  not  exist ;  hence  we  say  that  the  total  area  between  the 
a;-axis  and  the  hyperbola  from  a;=  1  to  a;  =  co  does  not  exist* 

*  This  is  the  standard  short  expression  to  denote  what  is  quite  obvious,— 
that  the  area  up  to  x  —  m  becomes  infinite  as  in  becomes  infinite.  This  result 
makes  any  consideration  of  tlie  area  up  to  ^  =  x  perfectly  useless  ;  hence  the 
expression  "  fails  to  exist,"  which  is  slightly  more  general. 


202 


GENERALIZED  INTEGRALS 


[VII,  §  111 


111.  Integrand  Infinite.  Vertical  Asymptotes.  If  the  func- 
tion to  be  integrated  becomes  infinite,  tl\e  situation  is  precisely 
similar  to  that  of  §  110 ;  grapliically,  the  curve  whose  area 
is  represented  by  tlie  integral  has  in  this  case  a  vertical 
asymptote. 

Itf(x)  becomes  infinite  at  one  of  the  limits  of  integration, 
jc  =  6,  we  define  the  integral,  as  in  §  110,  by  a  limit  process : 

Xx=l>  /•6-c 

f(x)  dx  =  \im.  I       f(x)  dx. 

A  similar  definition  applies  if  f{x)  becomes  infinite  at  the 
lower  limit,  as  in  the  following  example. 

Exam2:)le  1.  The  area  between  the  curve  y  =  l/-\/x  and  the 
two  axes,  from  a;  =  0  to  cc  =  1,  is 

^1  ^^  =       r^'—^dx  =  limf  r^'— =da;1 
J  ^=0         *^x=o   Va;  ^^  [yx=c   -Vx     J 

=  limr2\/^1"    =limr2-2Vc     =2. 


y 

V 

= 

X- 

i 

\ 

^\ 

- 

'y/\. 

^/^z 

s 

y^. 

__ 

0  c 

X 

Example  2.  The  area  be- 
tween the  hyperbola  y  =  1/x, 
the  vertical  line  x  =  1,  and  the 
two  axes,  does  not  exist.     For, 

I      -dx==  log  x\      =  —  log  c, 

Jx=c     X  Jx=c 


but  lini  (—  log  c)  as  c  =  0  does 
not  exist,  for  —  log  c  becomes 
infinite  as  c  =  0. 


VII,  §  112] 


IMPROPER  INTEGRALS 


203 


Examples.     The  area  between  the  curve  y  =  l/-^x—i,  its 
asymptote  x  =  1,  and  the  line  x  =  2  is 


f --^^  =  lim  r-^^  =  ?lim(l_c^/3)=3. 

Jl      ^X-1  c^uJl+c^/x-1        ^e=y,  ^2 

112.  Precautions.  It  is  dangerous  to  apply  limits  of  in- 
tegration between  which  the  integrand  becomes  inhnite  or  is 
otherwise  discontinuous. 

Example  1.      Show  that   I      1/x^dx  does  not   exist.      The 

ordinate  y  =  1/xr  becomes  infinite  as  x  approaches  zero,  i.e.  the 
y-axis  is  a  vertical  asymptote.  Hence  to  find  the  given 
integral  we  must  proceed  as  in  §  111,  breaking  the  original 
integral  into  two  parts : 


Jx=c  ar  x_\x=c     c 

ii=f-i..=-il~-=i-i. 

*^'x=-i  or  a;Jx=-i     c 


^    I 

t\ 

>y. 

k; 

\ 

>'<>. 

:^ 

V/A 

|f^ 

-4 

■ 

/: 

\ 

y 

\     y 

= 

7 

'i 

/ 

Vj 

- 

— 

-^ 

- 

1 

J 

1 

The  limit  of  neither  exists  since 
1/c  becomes  infinite  as  c  =  0; 
hence  the  given  integral  does 
not  exist. 

Carelessness  in  such  cases  re- 
sults in  absurdly  false  answers ;  thus  if  no  attention  were  paid 
to  the  nature  of  the  curve,  some  person  might  write : 


Fig.  40 


Jz=-1        Jx=-\    X-  ^  '      X    J^_i 


(sic!)- 1-1  = 


which  is  ridiculous  (see  Fig.  46). 

The  only  general  rule  is  to  follow  the  principles  of  §§  110- 
111  in  all  cases  of  infinite  limits  or  discontinuous  integrands. 
Such  integrals  are  called  improper  integrals. 


204  GENERALIZED  INTEGRALS  [VII,  §  112 

EXERCISES  XLIV.- IMPROPER  INTEGRALS 

"Verify  the  following  results  : 

.    ri  dx  .    j  determinate  if  n  <  1, 
Jo  a?*        \  non-existent  if  n  ^  1. 


)    C"      ^^      =2\/a.  (o)    P      ^^      is  non-existent, 

-^o  V^^^  A52X-3 

M    f^        dx 
^  Ji.5(2x-3)» 


y/a- 
*2        f^x         ,■„    f  determinate  if  n  <  1, 
non-existent  if  n  ^  1. 
•i        dx 


State  a  similar  rule  for 


f 


{hx  +  A:)" 

f"  rfx  ^^.  (;;.)      r_     ^^^^         =§Zg!. 

Jo  V^i^TT^      2  '   '   Jo  V^^^:^         8 

.      C     ^dx    _^  n)    C+^ dx is  non-existent. 

^  Jo   vr^^2  J   1  x'^  +  5  X  -t-  4 

2.  Show    that    the    integrals     j '  tan  x  dx,   j'cotxdx,    J    secxdx, 
Jo        X 

3.  Verify  each  of  the  following  results  :  j 

^^    rdx     1  /„N     p        t?x        _  g 

")   ji    x^  =  2-  ^^    Jo    (l  +  x)3/2-2- 

6)     r    —  is  non-existent.  (-f^)    C^  — '^ —  is  non-existent. 

^     Jl      ^  ^     ^     Jo     (l-fx)2/3 


r"       dx        •      f  determinate  if  n  >  1, 
Jo    (1 -t- x)"        I  non-existent  if  71 -^  1. 

^'  rrf;;r»=r  ®  I, 

.        f"  dx  IT  /.oo 

^^     Ja     ^M^  =  4^*  0)      jo     ^-^^^^  = 


~  i?  dx  is  non-existent 

X 


non-existent. 


VII,  §  112]  IMPROPER  INTEGRALS  205 

4.  Determine  the  a,rea  between  each  of  the  following  curves,  the  x- 
axis,  and  the  ordinates  at  the  values  of  x  indicated  : 

(a)  2/3(x  _  1)2  =  1 ;  X  =  0  to  9.  Ans.  9. 

(6)  xy-{\  +  .<■)-  =  4  ;  x  =  0  to  4.  Ans.  4  tan-i  2. 

(c)  ?/'-x*(l  +  x)  =  1 ;  X  =  0  to  3.  ^ns.  qo. 

((?)  x22/2(x2  -  1)  =  9  ;  X  =  1  to  2.  Ans.  2  jr. 

(e)  2/3(x-l)'-  =  8x3;  x  =  0to3.  Ans.  9S/2  +  9/2. 

(/)  x2r/2(x2  +  9)  =  1 ;  X  =  4  to  ».  Ans.  %  log  2. 

(</)  2/"(l  +  t)-*  =  X  ;  X  =  0  to  00.  Ans.  ir. 

(A)  2/3(x  + 1)-  =  1  ;  X  =  0  to  00.  Ans.  oo. 

5.  If  each  of  two  curves  y  =/(x)  and  y  =  4>(x)  is  asymptotic  to  the 
y-axis,  and  if  /(x)  >  0(a;)  ^  0,  show  that  J  /(x)(?x  cannot  exist  unless 
^^(f>{x)dx  exists.  Hence  show  that  J^x-2rfx  does  not  exist  by  comparing 
it  with  jjx-i  dx. 

6.  If  each  of  two  curves  y  =/(x)  and  y  =  <^(x)  is  asymptotic  to  the 
X-axis,  and  if  /(x)  >  0(x)  ^  0,  show  that  J  f(x)dx  cannot  exist  unless 
J^</)(x)  dx  exists. 

7.*  Verify  each  of  the  following  results  (see  Tables,  IV,  F  and  V,  F)  : 

x*e~*  dx  =  4  ! 

r"e--'  dx  =  n  ! 

8.*  Show  by  means  of  Exs.  6  and  7  that  jjxi-^e-' dx  exists.  Find  its 
value  from  the  tables  (IV,  F,  109  and  V,  F).  Find  the  value  of 
Qx--^e-^  dx  ;    the  value  of  j"x^-^e-*  dx  ■  the  value  of  J^  3^*6"^  dx. 

9.  If  x"  ^/(x)  ^  0  for  large  values  of  x,  where  n  is  a  positive  integer, 
show  that  f^  f{x)e-'  dx  exists.  Hence  show  that  e'  >  x"  for  large  values 
of  X  by  showing  that  J*e'  ■  e-'  dx  does  not  exist. 


(a)  j'^'-dx  =  l.        (c)    j|"x2e-dx  =  2. 

(^)  r 

(ft)    P  .re-'  dx  =  1.     (d)     C^^<i-'  dx  =  3  ! 

(/).  f: 

206  GENERALIZED  INTEGRALS  [VII,  §  113 

113.  Repeated  Integration.  Repeated  integrations  may  be 
performed  with  no  new  principles.     Thus 

i  --dx= he;       and      i( \-c\clx= —  log x  + ex  +  c'. 

The  final  answer  might  be  called  the  second  integral  of  1/a^. 
Such  processes  are  frequently  used  in  solving  differential 
equations  (see  §  92,  and  Chapter  X). 

Thus,  in  the  case  of  a  falling  body,  the  tangential  acceleration  is 
constant : 

jT--7:=  -g, 
dt 

where  g  is  the  constant ;  hence 

V  =  Ijfdt  +  const.  =  —  gt  +  c ; 
but  since  v  =  ds/dt, 

s  —  jvdt  4-  const.  =  j  (—  gt  +  c)  dt  +  const.  =  —  ^  +  c<  +  c'. 

If  the  body  falls  from  a  height  of  100  ft.,  with  an  initial  speed  zero, 
s  =  100  and  v  =  0  when  t  =  0;  hence  c  =  0  and  c'  =  100,  whence  we 
tiud  s  =  -  gty2  +  100. 

The  equations  s=  1^^^  + const.,  v  =  J.;Vf^^  +  const.,  just  ob- 
tained, apply  in  any  motion  problem,  where  jj.  is  the  tangential 
acceleration,  v  is  the  speed,  and  s  is  the  distance  passed  over. 
Substituting  for  v,  we  might  write 


s  =  |[j'jV(^«  +  c]di  +  c'. 


114.  Successive  Integration  in  Two  Letters.  Another  dis- 
tinctly different  case  of  repeated  integration  which  can  be 
performed  without  further  rules  is  that  in  which  the  second 
integration  is  performed  with  respect  to  a  different  letter. 

Thus,  the  volume  of  any  solid  is  (§  70,  p.  121), 


(1)  ^=£ 

where  As  is  the  area  of  a  section  perpendicular  to  the  direc- 


As  dh, 

'h=a 


VII,  §  114]  SUCCESSIVE  INTEGRATION 


207 


60,   p.  103,   and  then  iu- 

z 


tion  in  which  h  is  measured,  and  where  /t  =  a  and  /i  =  6  de- 
note planes  which  bound  the  solid. 

In  many  cases  it  is  convenient  first  to  find  ^-l^  by  a  first 
integration,   by   the   methods   of 
tegrate  Ag  to  find    V  by  (1), 
this  second  integration  being 
with  respect  to  the  height  h. 

Example  1 .     Find  the  volume  of 
the  parabolic  wedge 

y^  =  xii  -  zy 

between  the  planes  s  =  0  and  z  =  1 
and  between  the  planes  x  =  0  and 
x  =  l. 

The   area  As  of    a    section   by 
any  plane  z  =  h  parallel  to  the  xy-plane  is  twice  the  area  between  the 
curve  y  =  (1  —  A)  Vx  and  the  x-axis : 


^^]i;=-X^'^^^-=2C(^-^>^'^-=|<i-'^>^^^]n 


x=i      4 


(1-A), 


hence  this  volume,  by  (1),  is 


""-A' 


2/Ja=o 


Notice  that  h,  during  the  first  integration,  was  essentially  constant. 
Notice  also  that  the  volume  of  the  wedge  is  one  third  the  volume  of 
the  circumscribed  rectangular  parallelopiped  ;  and  that  since  ^5  is  a 
linear  function  of  A,  the  prismoid  rule  (§  71,  p.  125)  gives  the  volume 
precisely. 

Combining  the  formulas  used  in  this  example,  the  volume 
Fmay  be  written 

V~\  '^'  =  C^'\2  (^^ (1  -  h)  V^  clAlh  =  ? . 

Ja=o       »/A=0    [^    ^x=0  J  3 

Such  successive  integrals  in  two  letters  are  very  common 
in  all  applications. 


208  GENERALIZED  INTEGRALS  [VII,  §  114 

EXERCISES  XLV.  —  SUCCESSIVE  INTEGRATION 

1.  Determine  a  function  y  =  f  (x)  whose  second  derivative  d-y/dx-  is 
6  X.     Alls.  y  =  xs+  Cix  +  Co. 

2.  Determine  tlie  speed  v  and  tlie  distance  s  passed  over  by  a  particle 
whose  tangential  acceleration  d-s/dt-  is  (3 1.  Find  the  values  of  the  arbi- 
trary constants  if  v=0  and  s=0  when  t=0;  if  v=:100  and  s=0  when  t=0. 

3.  Find  the  general  expressions  for  functions  whose  derivatives  have 
the  following  values : 

(a)  d'-y/dx^  =  6  x^.        (d)  d'^r/dd^  =  1/ VI^.     (p-)  d^y/dx'-^  =  e^ 
(6)  d2s/d«2  =  1  +  2  «.     (e)  dV/d^s  =  ^2  _  2  ^.       (/i)  d'^s/f^^"  =  sec^  t. 
(c)  d2s/d«2=Vl-«.     (/)  d%/rfw3  =  1  +  M'-2.       (j)  d^dir^  =  l/w'-. 

4.  Determine  the  speed  v  and  distance  s  passed  over  in  time  t,  when 
the  tangential  acceleration  jy  and  initial  conditions  are  as  below  : 

(a)  Jt  =  sin  ^ ;  w  =  0  and  s  =  0  when  «  =  0, 

(6)  jj.—  t  +  cos ^ ;  ■?;  =  0  and  s  =  0  when  «  =  0. 

(c)  jj.  =  VT+1 ;  V  =  3  and  s  =  0  when  «  =  0. 

(d)  ^V  =  ^/Vl  +  t^  ;  V  =  1  and  s  =  0  when  t  =  0. 

5.  Evaluate  each  of  the  following  integrals,  taking  the  inner  integral 
sign  with  the  inner  differential : 

(h)    \        \       Gx^{l-y)dydx.  (g)    \  \  {x+y)dydx. 

r-  p=3  (^2^1)  (4-2/2)d2/  dx.     (/t)    r='  r'=''^'  (X  +  ?/)'^  dydx. 

Jx=0    Jy=2  Jx=0   Jy=\ 

(d)    r~    r^  Vrt  +  u  du  dv.  (j)     ^  P''  p"'"^(x  +  ?/+2)d2  di/d.r. 

6.  Find  the  volume  of  the  part  of  the  elliptic  paraboloid  4  x^  +  O  y'^-Zd  z 
between  the  planes  z  =  0  and  z  =  \;  between  the  planes  z  =  a  and  z  =.b. 


VII,  §114]  SUCCESSIVE  INTEGRATION  209 

7.  Find  the  volume  of  the  part  of  the  cone  4  x-  +  9  y-  =  36  s-  between 
the  planes  2  =  0  and  z  =  2  ;  between  z  =  a  and  z  =  b. 

8.  Find  the  volume  of  the  part  of  the  cylinder  x"^  +  y^  =  25  between  the 
planes  z  =  0  and  z  =  x  ;  between  the  planes  z  =  x/2  and  2  =  2  x. 

9.  A  parabola,  in  a  plane  perpendicular  to  the  x-axis  and  with  its  axis 
parallel  to  the  2-axis,  moves  with  its  vertex  along  the  x-axis.  Its  latus  rectum 
is  always  equal  to  the  x-coordinate  of  the  vertex.  Find  the  volume  inclosed 
by  the  surface  so  generated,  from  2  =  0  to  2  =  1  and  from  x  =  0  to  x  =  1. 

10.  Find  the  volume  of  the  part  of  the  cylinder  x'^  +  2/2  =  9  lying  within 
the  sphere  x^  +  y-  +  z-  =  16. 

11.  For  a  beam  of  constant  strength  the  deflection  y  is  given  by  the 
fact  that  the  flexion  is  constant :  b  =  d-y/dx-  =  const,  if  the  beam  is  of 
uniform  thickness.  Find  y  in  terms  of  x  and  determine  the  arbitrary  con- 
stants if  y  =  0  when  x  =  ±  1/2.  [This  will  occur  If  the  beam  is  of  length  1, 
and  is  supported  freely  at  both  ends.] 

12.  Determine  the  arbitrary  constants  in  the  case  of  the  beam  of  Ex. 
11,  if  y  =  0  and  dy/dx  =  0  when  x  =  0.  [This  will  occur  if  the  beam  is 
rigidly  embedded  at  one  end.] 

13.  For  a  beam  of  uniform  cross  section  loaded  at  one  end  and  rigidly 
embedded  at  the  other,  b  =  d^y/dx^  =  k{l  —  x)  where  I  is  the  length  of  the 
beam,  x  is  the  distance  from  one  end,  and  A;  is  a  known  constant  which  is 
determined  by  the  load  and  the  cross  section  of  the  beam.  Find  y  in 
terms  of  x,  and  determine  the  arbitrary  constants. 

14.  Find  y  in  terms  of  x  in  each  of  the  following  cases  : 

(a)  d^y/dx:^  =  Jc{r^  -  2  Ix  +  x'^);  y  =  0,  dy/dx  =  0  wiien  x  =  0. 
[Beam  rigidly  embedded  at  one  end,  loaded  uniformly.] 
(6)  d^y/dx^  =  a  +  bx;  y  =  0,  dy/dx  =  0  when  x  =  0. 
[Beam  of  uniform  strength  of  thickness  proportional  to  (a  +  6x)-i,  em- 
bedded at  one  end.] 

(c)  d^y/dx-  =  ^•(^V8  -  x'^/2) ;  y  =  0  when  x  =  ±  1/2. 
[Beam  supported  at  both  ends,  loaded  uniformly.] 

(d)  dhj/dx-  =  ^•/x2  ;    2/  =  0,  dy/dx  =  0  at  x  =  ?. 

[Beam  of  uniform  strength  of  thickness  proportional  to  x^,  embedded  at 

15.  Find  the  angular  speed  w  and  the  total  angle  d  through  which  a 
wheel  turns  in  time  ^,  if  the  angular  acceleration  is  a  =  dr0/dfi  =  2t, 
and  if  ^  =  w  =  0  when  t  =  0. 


210 


GENERALIZED   INTEGRALS 


[VII,  §  115 


(1) 


Fig.  48 


115.  Double  Integrals.  It  is 
often  convenient  to  restate  such 
problems  as  that  solved  in  §  114 
in  somewhat  different  form. 

In  obtaining  "the  area  Ag  we 
originally  (§  66,  p.  115)  cut  the 
area  into  strips  of  width  Aa;; 
their  length  is  2  y  each,  since 
they  reach  from  one  side  of  the 
parabola  to  the  other.  We  then 
showed  that 


As=2  \        y  doc  =  2 lim    [Sum  of  terms  like  y  Ace.] 

Jx=0  Aa;=0 


We  may  as  well  proceed  to  set  up  both  integrations  at  once, 
as  follows  :  let  us  consider  the  small  column  whose  face  is  2 1/  Ax 
and  whose  thickness  is  Ah ;  its  volume  is 

(2)  2yAxA7i; 

the  volume  of  the  whole  layer  whose  base  is  As  is 

(3)  As-  Ah  =  2  Ah  .   r~\j  dx  =2Ah-  lim  T"=^  (y  Ax), 

where  2  stands  for  "  the  sum  of  terms  like  " ;  hence 

(4)  As  -Ah  =  2  lim  Ah  ■  V'='  (yAx)  =  2  lim  V  ^'  (y  Ax  Ah). 

The  entire  volume  is,  however  (§  70,  p.  121) : 

(5)  Fl  *^'  =  f^'Asdh  =  lim  V'^AsAh 

=  2  lim  y*=ilimy^\ 


Ax  Ah). 


VII,  §116]  DOUBLE  INTEGRALS  211 

The  expression 


which  occurs  in  (o)  is  equal  to  the  double  limit  which  follows ; 
it  is  called  a  double  integral,  and  is  denoted  by  \\ydxdh: 


(«)     is2;:::2::^^-^'-Xrxr* 


dx  dh. 


In  the  particular  example  in  hand,  y  =  (1  —  ^)Vic,  and  the 
limits  are  (h  =  0,  h  =  l)  and  (x  =  0,  x  =  l);  but  the  argument 
was  not  affected  by  our  knowledge  of  these  values.  It  follows 
that  the  successive  integrals  mentioned  in  §  114  are  always 
equal  to  the  double  limit  in  (6),  where  y  is  any  function  F(x,  h) 
of  X  and  h  we  please : 

(7)    f^r  r"^''i^(x,/i)ri^1rZ7i  =  limT';:^  V^:%A»Afc 

I  Fix,  h)  dx  dh. 

h  =  a  Jx  =  c 

This  is  the  fundamental  summation  formula  for  double  in- 
tegrals. In  writing  it,  any  letters,  not  necessarily  x  and  /i,  may 
be  used.  Moreover  c  and  d  may  depend  on  /i,  as  we  shall  see 
in  numerous  examples.  It  is  used  exactly  as  we  have  used  the 
original  summation  formula :  quantities  we  desire  to  measure 
often  appear  most  naturally  in  the  forms  of  approximate 
double  sums  like  (6).  The  accurate  evaluation  is  done  by  suc- 
cessive integration  by  means  of  (7). 

116.  Illustrative  Examples.  In  this  paragraph,  several 
applications  of  double  integration  are  worked  out.  These 
should  not  be  memorized,  but  rather  the  formulas  should  be 
built  up  by  the  student  each  time  they  are  used. 


212 


GENERALIZED   INTEGRALS  [VII,  §  116 


[A]   Volumes  by  Double  Integration.*    A  problem  which  is  essen- 
tially the  same  as  that  of  §  114  is  to  find  the  volume  under  any  surface 

whose  equation  is  given  in  the  form 
z  =  F(x,  ?/),  where  F(x,  y)  is  any 
function  of  x  and  y.  Consider  for 
example  the  volume  V  bounded  by 
the  surface,  a;5;-plane,  the  planes 
x=a  &nd  x=h  and  the  right  cylin- 
der whose  base  is  a  given  curve 
y  =f(x)  of  the  a:2/-plane. 

If  the  volume  is  divided  into 
layers  by  planes  parallel  to  the 
ys-plane,  equally  placed  at  intervals 
Ax  ;  and  if  these  layers  are  them- 
selves divided  into  small  columns 
of  width  Ay,  the  volume  of  any  one 
column  is  approximately  z  Ay  Ax, 
and  the  total  volume  is 


lira  '%-^  = 


Ay  Ax 


=cz 


F(x,  y)dydx. 


Thus  the  volume  under  the  surface  z  =  x^  +  y"  between  the  a-^-pkne, 
the  planes  x  =  0  and  x  =  1,  and  the  cylinder  whose  base  is  y  =  Vx  is 


Jx=0  Jy= 


(x2  +  y^)  dy  dx 


x:'[^ 


•■y  +  y-\        dx 
3  J»=o 


=£;■(-- 


105* 


[JB]   Area  in  Polar  Coordinates. 

The  area  A  bounded  by  a  curve  whose 
equation  in  polar  coordinates  is  p=f{d), 
and  two  radii  vectors  6  =  a,  d  =  ^  is 
approximated  by  dividing  it  into  trian- 
gular strips  by  radii  vectors  spaced  at 
equal  angles  A^.  If  we  then  draw  cir- 
cles with  centers  at  0,  equally  spaced 
at  intervals  Ap,  the  whole  area  A  is 
divided  into  small  curvilinear  "squares"  like  the  one  shaded  in  ^ig.  50. 

*  Formulas  from  Solid  Analytic  Geometry  are  to  be  found  in  Chapter  IX. 


VII,  §  116] 


DOUBLE  INTEGRALS 


213 


The  straight  line  side  of  one  of  these  is  Ap,  while  the  circular  side  has  a 
length  pAe,  where  p  is  the  value  of  p  along  that  side.  Hence  the  area  of 
the  shaded  "  square  "  is,  approximatelj^  pApAd  and  the  area  to  be  found 
is,  precisely, 


lim 


xzx 


p=/(e) 


pApAe 


\        \  p  dp  dd. 

Je=^a  Jp=0 


The  first  integration  Jp  dp  can  always  be  performed,  since  I pdp=p-/2; 
but  it  is  best  not  to  burden  the  memory  *  with  this,  since  it  is  evident  each 
time  such  an  area  is  to  be  found.  Thus  the  area  bounded  by  the  curve 
p  =  sec  6  (draw  it)  and  the  lines  d  =  0,  6  =  7r/4  is 


Je=o 


x: 


p  dp  dd  : 


Je=o      L2jp=o 

1  n  e=T/4      1 
-  tan  6'  =    . 

2  Jd=i>        2 


Je=o 


dd 


[C]  Moment  of  Inertia  of  a  Thin  Plate.  The  moment  of  inertia 
J  about  a  point  O  of  a  small  object  whose  mass  is  m  is  defined  in 
Physics  to  be  the  product  of  the  mass 
times  the  square  of  the  distance  from 
O  to  the  object :  /  =  mr'^. 

Given  now  a  thin  plate  of  metal  of 
uniform  density  and  thickness,  whose 
boundary  C  is  a  given  curve,  let  us 
divide  the  plate  into  small  squares 
by  lines  equally  spaced  parallel  to  two 
rectangular  axes  through  0.  Let  P  be 
a  point  in  any  one  of  these  squares 
and  let  0P  =  r  =  Vx^  ■+  yK  Then  the 
mass  of  the  square  is  k  •  Ay  Ax  where 
k  denotes  the  constant  surface  density 
(i.e.  the  mass  per  square  unit) ;  and  the 

moment  of  inertia  of  this  square  about  0  is,  approximately,  k 
Hence  the  moment  of  inertia  /  of  the  entire  plate  about  0  is  : 


1, 

^"^ 

■^ 

/ 

'     ^I- 

i 

/ 

V 

A/ 

'            - 

'?;'; 

\ 

/ 

,  / 

J 

\      / 

y 

y-^  ; 

__    ^ 

0 

/ 

V 

- 

-Ax^              X 

Fig.  51 


r-  Ay  Ax. 


lim 


i=^>^XX^''^-'^y^Si 


fjf-  +  y-)  dy  dx. 


*  If  any  part  of  this  work  is  memorized,  it  should  be  at  most  the  figure 
drawn  above. 


214  GENERALIZED  INTEGRALS  [VII,  §  116 

where  proper  limits  of  integration  are  to  be  inserted  to  cover  the  area  en. 
closed  by  C.  If  G  is  an  oval,  as  shov?n  in  the  figure,  the  limits  of  y  are 
the  values  of  y  along  the  lower  half  oval  and  the  upper  half  ;  these  must 
be  given  in  the  problem  as  functions  of  x.  The  limits  for  x  are  the  ex- 
treme values  of  x  on  the  two  ends  of  the  oval. 

Thus  the  moment  of  inertia  of  a  plate  bounded  by  the  two  curves  y  = 
(1  —  x'^)  and  2/  =  (a:2  —  1),  about  the  origin  (draw  the  figure)  is : 

I=k  r^^^  Cy-='^-^'  .2  ^  2/2)d2/  dx  =  k  f'^^Yx^?/  +  ^  T"^~"'dx 

Ji=-1    Jy=x2-1  Ji=-1    L  3  _\y=x2-l 

=  l^C=^\i-x^)ax  =  mx-^r^'  =  ^-k, 

3  Jx=-i  ^  ^  3    1  7jx=-i      7    ' 


where  k  is  the  surface  density. 

[D]  Moment  of  Inertia  in  Polar  Coordinates.  Using  the  figure 
drawn  for  [B],  it  is  easy  to  see  that  the  moment  of  inertia  of  a  thin  plate 
of  the  shape  of  the  area  in  [B]  is  : 

/ = lim  k .  y'-'  y'^''''  p^ApM  =  k.  r'  r''''  p'  ^p  ^^' 

A8^ 

where  A;  is  the  surface  density  {i.e.  mass  per  unit  area)  as  in  [C]. 

Thus  for  a  circle  whose  center  is  O,  p=f{e)  =  a,  the  radius.  Hence, 
the  moment  of  inertia  of  a  circular  disk  about  its  center  is  : 

i=k.r'''[^T'^cie=k-r'''-cio=.k^=.^, 

Je=o    L4jp=o  Jfl=o     4  2         2 

where  k  is  the  surface  density,  and  M  =  kira^  is  the  mass  of  the  disk. 

EXERCISES   XL VI.— DOUBLE   INTEGRALS 

1.  Find  the  volume  under  the  surface  z  =  x'^  +  y'^  between  the  xz- 
plane,  the  planes  x  =  0  and  x  =  1,  and  the  cylinder  whose  base  is  the 
curve  y  —  x^. 

2.  Find  the  volume  between  the  xy-plane  and  each  of  the  following 
surfaces  cut  off  by  the  planes  and  surfaces  mentioned  in  each  case  : 

(a)   z  =  x  +  y  cut  off  by  ?/  —  0,  X  =  0,  X  =  1,  2/  =  Vx. 

(&)    z  =  x^  +  y        cut  off  by  ?/  =  0,  X  ==  1,  X  —  3,  2/  =  x^. 
(c)    z  =  xy  cut  off  by  2/  =  0,  X  =  2,  X  =  4,  ?/  =  x^  -}- 1. 


VII,  §  116]  DOUBLE  INTEGRALS  215 

cut  off  by  J/  =  0,  a;  =  1,  a;  =  5,  y  =r  a^. 
cut  ofl  by  y  =  0,  X  =  0,  X  =  1,  y  =  x^. 
cut  off  by  X  =  0,  y  =  1,  y  =  4,  y'^  =  x. 
cut  off  by  a;  =  0,  y  =  2,  2/  =  5,  y  =  X. 
cut  off  by  y  =  0  and  y  =  1  —  x^. 
cut  off  by  y  =  x'^  and  y  =  1. 
cut  off  by  y  =  x^  and  y  =  x. 
cut  off  by  y  =  x-  and  y  =  2  —  x^. 

3.  Find  the  volume  of  the  portion  of  the  paraboloid  z  =  1  —  x^  —  4  y^ 
which  lies  in  the  first  octant. 

4.  If  two  plane  cuts  are  made  to  the  same  point  in  the  center  of  a 
sircular  cylindrical  log,  one  perpendicular  to  the  axis  and  the  other  mak- 
ing an  angle  of  45°  with  it,  what  is  the  volume  of  the  wedge  cut  out  ? 

5.  Show  that  the  volume  common  to  two  equal  cylinders  of  radius  a 
which  intersect  centrally  at  right  angles  is  16  a^/3. 

6.  Show  that  the  volume  of  the  ellipsoid  x716  +  yV9  +  274  =  1  is  32  tt. 

7.  What  part  of  the  ellipsoid  in  Ex.  6  lies  within  a  cube  whose  center 
is  at  the  origin  and  whose  edges  are  6  units  long  and  parallel  to  the 
3oordinate  axes  ? 

8.  AVhere  should  a  plane  perpendicular  to  the  x-axis  be  drawn  so  as 
to  divide  the  volume  of  the  ellipsoid  in  Ex.  6  in  the  ratio  2:1? 

9.  Calculate  by  double  integration  the  areas  bounded  by  the  following 
Burves : 

(a)    y  =  X-  and  y  =  Vx.  (e)    x  =  0,  y  =  sinx,  and  y  =  cos  x. 

•    (5)    y  =  x2  and  y  =  x^.  (/)  y  =  0,  y2  —  x.,  and  x^  —  y-  =  2. 

(c)  y  =  x2  and  -  x^  +  y2  =  2.  (g)    y  =  2x,  y  =  0,  and  y  =  1  —  x. 

(d)  x2  +  y'^  =  12  and  y  =  x'^.  (A)  y2  =  x,  and  y  =  1  —  x. 

10.  Calculate  the  moment  of  inertia  of  a  thin  plate  bounded  by  the 
curves  y  =  x^,  y  =  2  —  x^^  about  the  origin. 

11.  Calculate  the  moment  of  inertia  of  a  thin  plate  about  the  origin, 
in  each  of  the  cases  in  which  the  shape  of  the  plate  is  the  area  bounded 
by  the  curves  in  one  of  the  parts  of  Ex.  9. 


216  GENERALIZED   INTEGRALS  [VII,  §  116 

12.  Find  the  moment  of  inertia  of  each  of  the  following  shapes  of  thin 
plate  : 

(a)  A  square  about  a  diagonal.     About  a  corner. 

(6)  A  right  triangle  about  a  side.     About  the  vertex  of  the  right  angle. 

(c)  A  circle  about  its  center. 

(d)  An  ellipse  about  either  axis.     About  the  center. 

(e)  A  circle  about  a  diameter. 

(/)  A  trapezoid  about  a  line  parallel  to  its  parallel  sides. 

13.  Find  the  moment  of  inertia  of  a  thin  spoke  of  a  wheel  about  the 
center  of  the  wheel. 

14.  Determine  the  entire  area,  or  the  specified  portion  of  the  area, 
bounded  by  each  of  the  following  curves,  whose  equations  are  given  in 
polar  coordinates  : 

(a)  p  —  2  cos  6.    Ans.  ir. 

(b)  One  loop  of  p  =  sin  2  0.    A7is.  ir/8. 

(c)  One  loop  of  p  =  sin  3  0.     Ans.  ir/l2. 

(d)  The  cardioid  p  =:  1  —  cos  ^.     Ans.  3  ir/2. 

(e)  The  lemniscate  p^  =  cos  2  0.     Ans.  1. 

(/)  The  spiral  p  =  0  from  ^  =  0  to  tt.     Ans.  ir^/6. 
(g)    The  spiral  p^  =  1  from  0  =  ir/i  to  ir/2.     Ans.  l/ir.  " 

(A)    p  =  1  +  2  cos  6  from  tf  =  0  to  tt.     Ans.  3  7r/2. 
(i)     p  =  tan  d  from  »  =  0  to  45°.     Ans.  1/2  -  tt/S. 
(j)    The  area  between  the  nth  and  (n  +  l)th  turns  of  each  of  the 
spirals  in  Exs.  14(/),  14  (gr). 

15.  Calculate  the  moment  of  inertia  of  a  thin  plate  about  the  origin, 
for  each  of  the  shapes  defined  by  the  areas  mentioned  in  Exs.  14  (a)-(O. 

16.  Calculate  the  following  moments  of  inertia  : 
(a)  A  thin  circular  plate,  about  its  center. 

(6)  A  thin  circular  plate,  about  a  point  on  the  circumference. 

(c)  A  thin  plate  bounded  by  two  concentric  circles,  about  the  center. 

(d)  An  equilateral  triangle,  about  its  center. 

(e)  An  equilateral  triangle,  about  one  vertex. 

17.  The  square  of  the  radius  of  gyration  p„  of  a  body,  about  any 
point,  is  its  moment  of  inertia  about  that  point  divided  by  its  mass : 
Pg^  =  T^  M.  Find  the  radius  of  gyration  for  the  example  solved  in  [C], 
§  110  ;  in  [Z)],  §  116. 

18.  Find  the  radius  of  gyration  for  each  of  the  thin  plates  described  in 
Exs.  9,  10,  12,  14,  16. 


Vll,  §  117]  MULTIPLE  INTEGRALS  217 

19.  n/(r,  y)  is  any  function  of  x  and  ij,  its  average  over  a  region  is 

Average  of  f{jc,  y)  =  j  j/Xx,  y)dxdy  -e- j  j  dxdy. 

Show  that  the  square  of  the  radius  of  gyration  about  the  origin  of  a 
ihin  plate  is  the  average  value  of  7^  =  x-  +  y'^  over  the  surface  of  the  plate. 

20.  Find  the  average  value  of  x  over  the  area  described  in  [C],  §  116. 
p.  214.     Find  the  average  value  of  y  over  the  same  area. 

[Note.  The  point  whose  coordinates  are  the  averages  of  values  oVx  and 
/  over  an  area  is  called  the  center  of  gravity  or  centroid  of  that  area.] 

21.  Find  the  centroids  of  each  of  the  areas  mentioned  in  Exs.  9  and  14. 

22.  Find,  for  the  area  mentioned  in  [C],  §  116,  p.  214,  the  average 
^alue  of  each  of  the  following  functions : 

(a)  xy.  (b)  a;2  +  4  y^.  (c)  x  +  y.  (d)  x^  -  y^. 

117.  Triple  and  Multiple  Integrals.  There  is  no  difficulty 
.n  extending  the  ideas  of  §§  113-llG  to  threefold  integrations 
)r  to  integrations  of  any  order.  Following  the  same  reason- 
.ng,  it  is  possible  to  show  that,  it  w  —  F{x,  y,  z) 

lim  V'^-'  ^y  y=d  yAr=«  ^^  ^^  ^^ 


--  f""  f^'  f^  F{x,  y,z)dxdy  clz, 

x/z=e     */y=c     »Jx=a 


svhere  the  three  integrations  are  to  be  carried  out  in  succes- 
sion, where  the  limits  for  x  may  depend  on  y  and  z,  and  where 
}lie  limits  for  y  may  depend  on  z :  but  the  limits  for  z  are,  of 
30urse,  constants. 

Thus  it  is  readily  seen  that  the  volume  mentioned  in  (^1), 
\  116,  may  be  computed  by  dividing  up  the  entire  volume  by 
;hree  sets  of  equally  spaced  planes  parallel  to  the  three  coor- 
linate  planes.  Then  the  total  volume  is,  approximately,  the 
sum  of  a  large  number  of  cubes,  the  volume  of  each  of  which 


218  GENERALIZED  INTEGRALS  [VII,  §  117 

is  Aa:  iyy  Az;  and  its  exact  value  is 


Az=y) 


Xx=5    /•.v=/(x)    /*z=F(x,xj) 
I  I  ds;  dy  dec, 

wliiph  reduces  to  the  result  of  \_A],  §  116,  if  we  note  that 

X^=F(x,  y)  '~\'=Fi,x,  y) 

Likewise  the  moment  of  inertia  /  (see  §  116,  [C])  of  the 
same  volume  with  respect  to  the  origin  is  approximately  the 
sum  of  terms  of  the  sort  h^x^  -\-  y^  -\-  z^)  Ax  A?/  Ag  where  k  is  the 
density  (mass  per  unit  volume) ;  whence  the  exact  value  of  /  is  * 


Xx=6    /».v=/(x)    /^z=F{=c,y) 
I  I  {j?  +  y'  +  z')dzdy 

=a  %/y=0         t/z=0 


dx. 


EXERCISES    XLVII.— MULTIPLE   INTEGRALS 

1.  Determine  the  volume  bounded  by  the  surface  z  =  {x  +  yy,  the 
coordinate  planes,  and  the  plane  x  ■\-  y  -{-  z  =  \. 

2.  Write  each  of  the  volumes  mentioned  in  Ex.  2,  and  in  Exs.  3-6, 
List  XLVI,  as  a  triple  integral ;  show  that  one  integration  reduces  the 
triple  integral  to  the  double  integral  used  before,  in  each  instance. 

3.  Find  the  volume  of  the  sphere  by  triple  integration. 

4.  Write  down  the  moment  of  inertia  about  the  origin  of  each  of  the 
solids  bounded  by  the  surfaces  mentioned  in  Ex.  2,  and  in  Exs.  3-6,  List 
XLVI.     Actually  carry  out  each  of  these  integrations. 

5.  Write  down  the  moment  of  inertia  of  a  right  cylinder  of  height  I 
whose  base  is  any  one  of  the  areas  mentioned  in  Ex.  9,  List  XLVI,  about 
an  axis  through  the  orighi  parallel  to  the  elements  of  the  cylinder.  Show 
that  one  integration  reduces  the  integral  essentially  to  the  double  integral 
used  in  List  XLVI,  in  each  instance. 

*  It  is  well  to  urge  that  such  formulas  should  not  be  remembered,  but 
obtained  in  each  exercise  by  the  simple  reasoning  used  above. 


^11,  §  118]     AVERAGES  — CENTERS  OF  GRAVITY        219 

6.  The  square  of  radius  of  gyration  of  a  solid  about  a  point  (or  about 
I  line)  is  the  moment  of  inertia  divided  by  the  total  mass.  Find  its  value 
:or  each  of  the  solids  mentioned  in  Ex.  4,  above  ;  for  each  of  the  figures 
nentioned  in  Ex.  12,  List  XLVI. 

7.  Find  the  average  value  : 

Average  of  /(x,  y,  z)  =  i  \  j/(x,  y,  z)  dxdydz  ^  j  j  idxdydz, 
)t  each  of  the  following  functions,  over  the  region  mentioned  in  Ex.  1 : 

(a)  /(«,  y,  3)  =  X.     (d)  f(x,  y,  z)  =  xyz.         (g)  /(.r,  y,  z)  =  x^  +  z^. 

(b)  f(x,y,z)=y.     (e)  f{x,  y,  z)  =  xy.  (h)  f(x,y,z)=  x'^ +  y^. 

(^c)  f{x,y,z)=z.    (/) /(x,y,3)  =  a;2+2/2.      (i)  f{x,y,  z)  =  x  +  y -]-z. 

[Note.  The  point  whose  coordinates  are  the  three  values  given  by  (a), 
6),  (c),  is  called  the  center  of  gravity,  or  centroid  of  the  volume.] 

8.  Find  the  centroid  of  the  solid  mentioned  in  Ex.  3,  List  XLVI. 

9.  Show  that,  in  spherical  coordinates  (p,  6,  0),  the  volume  of  a  solid 
3  given  by  an  integral  of  the  form  |  |  \  p^  sin  6  dd  d<p  dp.  where  0  is 
he  colatitude,  and  ^  is  the  longitude,  on  a  sphere  of  radius  p. 

10.  Calculate  the  volume  of  a  sphere  by  the  integral  in  Ex.  9. 

11.  Calculate  the  volume  cut  from  a  circular  cone  by  two  concentric 
pheres  with  centers  at  the  vertex  of  the  cone. 

12.  Show  that,  in  cylindrical  coordinates  (p,  0,  z),  the  volume  of  a 
olid  is  given  by  an  integral  of  the  form  |  i  \  pdddp  dz. 

13.  Calculate  the  volume  of  a  sphere  in  cylindrical  coordinates. 

14.  Determine  the  part  of  the  cylinder  p  =  2  sin  ^  which  lies  between 
he  planes  z  =  0  and  z  =  y. 

15.  Determine  the  part  of  the  cylinder  p  =  sin  2  ^  which  lies  between 
he  planes  z  =  0  and  x  +  y  +  z  =  V'2. 

118.  Other  Applications  of  Integration.  Averages.  Centers 
if  Gravity.  Among  other  ap])lifations  of  integration  already 
aentioned  in  exercises,  one  very  general  idea  is  that  of  the 


220  GENERALIZED  INTEGRALS  [VII,  §  118 

average  value  (A.  V.)  of  a  quantity : 


dx 
dx 


and  analogous  forms  for  functions  defined  in  a  given  area  or 
in  a  given  volume.  See  §  71,  p.  126 ;  Ex.  19,  p.  217 ;  Ex.  7, 
p.  219 ;  and  Tables,  IV,  H,  138. 

In  particular,  the  center  of  gravity,  or  centroid,  of  an  object 
is  defined  as  the  point  such  that  each  coordinate  of  that  point 
is  the  average  ot  the  same  coordinate  throughout  the  body ; 
thus,  for  a  thin  plate,  the  coordinates  (x,  y)  of  the  center  of 
gravity  are 

j  \  kx  dx  dy       \  Kkxdx  dy 
\  \  kdxdy  M 

j"  ^ky  dxdy      J  J  ky  dx  dy 


y  = 


dxdv  M 


where  k  is  the  surface  density  and  M  is  the  total  mass ;  and 
where  the  limits  of  integration  to  be  inserted  are  the  same  as 
those  inserted  in  finding  the  area  of  the  plate.  These  formulas 
hold  even  when  the  density  k  is  variable. 

Similar  formulas  hold  for  centers  of  gravity  of  solids ;  for 
other  averages,  such  as  the  center  of  water  pressure  on  a  dam ; 
and  for  a  variety  of  other  scientific  problems.  A  short  list  of 
these  formulas  is  given  in  the  Tables,  IV,  H.  These  may  be 
used  in  solving  exercises  which  follow. 


I 


II,  §  118]  GENERAL   EXERCISES  221 

EXERCISES  XLVIII.— GENERAL  PROBLEMS  m  INTEGRATION 
[These  problems  may  be  used  for  further  drill  and  for  re- 
iews;  it  is  advised  that  not  all  of  them  be  done  on  first  read- 
Dg.     Many  of  these  may  be  reserved  until  after  Chapter  IX.] 

1.  Carry  out  the  following  integratious  : 

a)  fH  +  f-ydt.  W  i-^,-  (0)  fe'Hsat. 

'   Ja  +  bz^  J  J     &\ne 

d)    (c+i'dz.  (k)    Csecx/2Unx/2dx.(r)    f —  "  ^^" 

J  •^  J  Vl  —  u'^  —  u* 

r  u  —  a  ^^^  ^j^     fcos«?tsm3«d?<.      {s)    f  (sec2a:  +  l)2tte. 

J  u  —  bifi  J  J 

•'  '  J  CO&26  ^       J  a- log  a; +1  ^  '    J 

.    C     xdx —  ,.    p2  +  4y_6^^  (tcos-^tdt. 

2.  Evaluate  the  following  definite  integrals  ;  notice  particularly  whether 
tie  integral  is  improper,  and  if  it  is,  explain  your  result : 

^    Jl    1  +4x2  J-1    Vl    -X2  -^O 

6)  j;;^e3^-Mx.  (^■)    r^^^"-  ^^^X^l  +  x2Ttan-ix- 

,>   Jo  (■?)  ji  VTT^  (a;+l)Vi^^ 

d)  j'_";cos2.xsinxdx.(^.)    r^^d».  (O   ^l"^^^- 

-''  Vm  —  2 

e)  I   a^logx-dx.  ^,v     p  <^^ (s)     I       xcosxcte. 

*^'  ^  -^    J«  at-  +  2bt  +  c'  ^" 

7)  |;;  (cos  2  x)2<fx.      ^^^  j'-cos2xcos3xdx.  ('^    roTl^' 


222  INTEGRATION  [VII,  §  118 

3.  By  use  of  numerical  tables,  find  the  values  of  the  following 
integrals : 

^  -'    Ji.43  X  +  1  -'1  -^-^   y/f^  +  9 

(g)    i  =•  (ft)    I      Vi-sin'-i^c?^. 

-^o     Vl  -  .36  sin2  0  ^       > 

«^30o  V5  -  2  sin2  a;  Jo      Vl^^  VT^^ 

4.  Show  that 

n dx 1  p dx <f> 

Jo  1  +  2  X  cos  <^  +  x2      2  Jo   1  +  2  X  COS  (^  +  x2      2  sin  0 ' 

and  explain  what  occurs  when  0=0,  and  when  ^  =  ir/2. 

5.  Integrate  the  following  general  integrals;  where /'(x)  denotes  the 
derivative  of /(x). 

(a)  j'/(x)/'(x)dx.  id)    jr(2x)dx. 

(&)  j"e*Ax)/'^x)dx.  («)    ij^^^' 

C  '  ff\    (    f'{x)dx 

(c)  J/'(cosx)sinxdx.  U)  J  i  +  [/(x)]2* 

6.  "Verify  the  result  of  integrating  sin^  x  (?x  by  comijaring  it  with  the 
integral  of  cos^  u  du  by  means  of  the  substitution  u  —  7r/2  —  x. 

7.  Evaluate  each  of  the  following  integrals : 

(a)    I        i       &\n{u-\-  v)dudv.  (6)    t       I       se'dtds. 

8.  Calculate  the  area  ^,  between  the  x-axis  and  the  curve  y  =  x^  —  9  x^ 
+  23  X  —  15,  from  x  =  1  to  x  =  3,  by  direct  integration  and  also  by  Simp- 
son's Kule.     Find  the  centroid  (x,  y)  of  the  same  area. 


VII,  §  118]  GENERAL  EXERCISES  223 

9.   Proceed  as  in  Ex.  8  for  each  of  the  following  curves,  between  the 
limits  stated  below : 

(a)  y  =  1  +  X  —  X-  +  x^  ;  x  =  0  to  a;  =  2. 

(b)  y  =  a(l  —  x-/b-);  1st  quadrant. 

Ans.  A  =  2b/S;  x  =  S  6/8,  y  =  2  a/5. 

(c)  y  =  x/(l  +  x^);  X  =  0  to  a;  =  1. 

Ans.   ^  =  (1/2)  loge  2;  x  =  0.6192,  y  =  0.2059. 

(d)  y  =  (e«  +  e-''x)/2  a  =  (1/a)  cosh  ax  ;  x  =  0  to  x  =  k: 

(e)  The  sine  curve  ;  one  arch.  Ans.  A  =  2  ;  x  =  7r/2,  y  =  tt/S. 
(/)  The  cycloid  ;  one  arch.  Ans.  A  =  3ira'^;  x  =  air,  y  =  5  a/0. 
(g)  a;2/3  +  j/2/3  -  a2/3[or  x  =  a  cos^  t,  y  =  a  sin^  t];  first  quadrant. 

^«s.    3  7raV32  ;  x  =  y  =  256  a/(315  n). 
Qi)  x  =  a  sin  i  +  ft  tan  t,  y  =  a  cos  < ;  <  =  0  to  «  =  ir/i. 

Ans.   A  =  a\v  +  2)/8  +  ab  log  tan  (.Stt/S). 
(i)  X  =  2  a  sin2  0,   j/  =  2  o  sin^  (/>  tan  (f>  ;    between    the   curve  and    its 
asymptote.    Ans.  A  =3 ncfi ;  x  =  b a/3 ;  j/  =  0. 

10.  Find  the  areas  bounded  by  each  of  the  following  curves,  or  the  part 
specified  : 

(a)  p  =  ad^;  one  turn.  (ft)  p  =  a  cos  d  +  b. 

(c)  p  =  a  sin  0  cos  ^/(sin^  ^  +  cos^  d)   [folium] ;  the  loop.     Ans.  a^JQ. 
{d)    ±x  +  Va2  -  y-  =  a  log  [(a  +  Va- -  y-)/?/]  [tractrix];  above  y  =  0. 
(e)  y"^(a  —  x)  =  x^  [cissoid] ;  to  its  asymptote  z  =  a. 

11.  Find  the  volume  generated  by  revolving  each  of  the  following  curves 
about  the  line  specified  : 

(a)  y  =  5x/(2 +  3x);  about  y  =  0;  X  =  0  tox=l.    Ans.  1.5558  ■••. 
(ft)  2  x2  +  5  y2  =  8  ;  about  y  =  0  ;  total  solid.     Ans.  64  7r/15. 

(c)  y"*  =  ax"  ;  about  y  =  0  ;  (0,  0)  to  (x,  y).     Atis.  mny^x/{2  n  +  m). 

(d)  y  =  b  sin  (x/a);  about  y  =  0;x  =  Otox  =  ir. 

(e)  y  =  a  cosh  (x/a);  about  y  =  0;  x  =  0  to  x  =  a. 
(/)  (■'■  —  «)^  +  2/'"  =  '''^  >  about  X  =  0  ;  total  solid. 

((j)  The  cycloid  ;  about  base  ;  one  arch.     Ans.  bir^a^. 

{h)  The  cycloid  ;  about  tangent  at  maximum  ;  one  arch.     Ans.  wa^. 

(i)  The  tractrix  ;  about  asymptote  ;  total.     Ans.  2  ira^/S. 

( j)  X  =  a  sin  «  +  ft  tan  t,  y  =  a  cos  t ;  about  y  =  0;  1  =  0  to  t  =  t. 

Ans.  7r[a'(sin  t  -  (1/3)  sin^  t)  +  a^ftf]. 

{k)  y"(a  —  x)  =  x^  ;  about  asymptote  ;  total  solid.     Ans.  2  v^a^. 

{I)  X  =  a  cos*  t,  y  =  a  sin*  t ;  about  y=0  ;  total  solid.     Ans.  32  wayiOb. 


224  INTEGRATION  [VII,  §  118 

12.  Obtain  a  formula  for  the  volume  of  a  spherical  segment  of  height  Ji. 

13.  Show  that  the  volume  of  an  ellipsoid  of  three  unequal  semiaxes, 
a,  6,  c,  is  4  7ra6c/3. 

14.  Show  that  the  volume  bounded  by  the  cylinder  a;2  +  y2  _  ^^x,  the 
paraboloid  x^  -\-y^=  bz,  and  the  xy-plane  is  (3/S2)('ira*/b). 

15.  Find  the  volume  common  to  a  sphere  and  a  cone  whose  vertex  lies 
on  the  surface  and  whose  axis  coincides  with  a  diameter  of  the  sphere. 

16.  Describe  the  solid  whose  volume  is  given  by  each  of  the  following 
integrals  ;  and  calculate  the  volume  : 

(a)    C  C^"'-^'  C^^'dzdydx.  (c)    f""  C  Cdzdydx. 

Jo  Jo  Jo  Jo    Jo  Jy 

(">  £1^"^'^' '''""•'■      (">  i'i/^  £"'■""■■■ 

17.  Show  that  the  a;-coordinate  of  the  center  of  gravity,  or  centroid, 
of  any  frustum  of  any  solid  is  : 

«=   (    x\    \   \dydz\dx-^V=    \   As-xdx-^  \    Asdx, 

if  As  is  the  area  of  a  section  perpendicular  to  the  x-axis,  Fis  the  total 
volume,  and  x  =  a  and  x  =  b  .are  the  truncating  planes.  State  similar 
formulas  for  y  and  z. 

[Note.     The  integral  (  (  ixdxdydz  is  often  called  the  moment  (or 
the  first  moment)  of  the  solid  about  the  yz  plane.] 

18.  Find  the  centroid  of  each  of  the  following  frusta :  * 

(a)  Of  the  paraboloid  x^  +  y^  =  4caz  by  the  plane  z  =  c.    Ans.  i  =  2  c/3. 
(6)  Of  a  hemisphere.     Ans.  z  =  S  r/8. 

(c)  Of  the  upper  half  of  the  ellipsoid  of  revolution  4x-  +  4y^  +  9z^  =  36. 

(d)  Of  the  upper  half  of  the  ellipsoid  x^  +  4y'^  +  9z-  =  36. 

(e)  Of  the  solid  of  revolution  formed  by  revolving  half  of  one  arch  of 
a  cycloid  about  its  base.     Ans.  x  =  ira/2  +  64  a/ (45  tt). 

19.  Show  that,  if  Ag  is  any  quadratic  function  of  x,  in  Ex.  17,  the 
moment  of  the  volume  about  the  yz  plane  is 

x-V=  [aB+bT+2{b-a)31^  (6  -  «)/6, 

where  B,  T,  M  denote,  respectively,  the  areas  of  cross  sections  by  a;  =  «, 
x~b,  x  =  (b  —  a)/2.     [Compare  §  71,  p.  126,  and  Ex.  3,  p.  128.] 


VII,  §  118]  GENERAL  EXERCISES  225 

20.  Find  the  lengths  of  the  arcs  of  each  of  the  following  curves,  between 
the  points  specified  : 

(rt)  y^logx;  x-atox  =  b.    Ans.   f  vT+x"-— log  {( vT+x2  +  l)/x}l   . 
(6)  e*  cosx  =  1  ;  a:  =  0  to  X  =  X.  -'" 

(c)  x^t^,  y  =  2at  (or  y-  =  4  a%) ;  t  =  tito  t  =  t2. 

Ans.  [tVa^  +  t~  +  a2  log  («  +  Va^  +  f^)V^ . 

(d)  One  arch  of  a  cycloid.     Ans.   8  a. 

(e)  p  =  a  (1  +  cos  6)  [cardioid]  ;  total  length.     Ans.   8  a. 

21.  Find  the  moment  of  inertia  and  the  radius  of  gyration  (Ex.  6,  List 
XLVII)  of  each  of  the  areas  mentioned  in  Ex.  9,  about  the  origin. 

22.  Calculate  the  moment  of  inertia  /  for  a  right  circular  cone  about 
its  axis.     Ans.    (3/10)  mass  •  square  of  radius. 

23.  Calculate  the  moment  of  inertia  and  the  radius  of  gyration  for  the 
rim  of  a  flywheel  about  its  axis,  the  inner  and  outer  radii  being  Bi,  i?2- 

Ans.   Mass  (iiJi^  +  i?2-)/2,  y/iB{^  +  Bi^)/2. 

24.  The  moment  of  inertia  of  an  ellipsoid  about  any  one  of  its  axes  is 
(1/5)  (mass)  (sum  of  the  squares  of  the  other  two  semi-axes). 

25.  Calculate  the  moment  of  inertia  for  a  spherical  segment  about  the 
axis  of  the  segment. 

26.  Show  that,  for  any  body,  2  /q  =  /-  +  /„  +  7^,  where  /o,  4,  /„,  I^ 
denote  respectively  its  moments  of  inertia  about  a  point  and  three  rectan- 
gular axes  through  that  point. 

27.  Show  that  for  any  figure  in  the  xy-plane,  I^—Ii  +  Iy,  where  /j,,  7„,  [^ 
denote  its  moments  of  inertia  about  the  three  coordinate  axes  respectively. 

28.  Show  that  the  total  pressure  on  a  rectangle  of  height  h  feet  and 
width  h  feet  immersed  vertically  in  water  so  that  its  upper  edge  is  a  feet 
below  the  surface  and  parallel  to  it,  is  G2.4  hh(a  +  h/2).  Show  that  the 
depth  of  the  center  of  pressure  is  at  (6  a-  +  6  ah  +  2  h^)/{Q  a  +  3h). 

29.  Show  that  the  total  pressure  on  a  circle  of  radius  r,  immersed 
vertically  in  water  so  that  its  center  is  at  a  depth  a+r,  is  62.4  irr'^{a+r). 
Show  that  the  depth  of  the  center  of  pressure  is  a  +  r  +  r^/  {i  r  +  4  a). 

30.  Show  that  the  total  pressure  on  a  semicircle,  immersed  vertically 
in  water  with  its  bounding  diameter  in  the  surface,  is  41.6  r^.  Show  that 
the  depth  of  the  center  of  pressure'  is  3  7rj'/16. 

31.  Show  that  if  a  triangle  is  immersed  in  a  liquid  with  its  plane  verti- 
cal and  one  side  in  the  surface,  the  center  of  pressure  is  at  the  middle  of 
the  median  drawn  to  the  lowest  vertex. 

Q 


226  INTEGRATION  [VII,  §  118 

32.  Show  that  if  a  triangle  is  immersed  in  a  liquid  with  its  plane  verti- 
cal and  one  vertex  in  the  surface,  the  opposite  side  being  parallel  to  the 
surface,  the  center  of  pressure  divides  the  median  drawn  from  the  highest 
vertex  in  the  ratio  3:1. 

33.  Calculate  the  mean  ordinate  of  one  arch  of  a  sine-curve.  The  mean 
square  ordinate.     [^Effective  E.  M.  F.  in  an  alternating  electric  current.] 

34.  Calculate  the  average  distance  of  the  points  of  a  square  from  one 
corner. 

35.  What  is  the  average  distance  of  the  points  of  a  semicircular  arc 
from  the  bounding  diameter  ? 

36.  When  a  liquid  flows  through  a  pipe  of  radius  E,  the  speed  of  flow 
at  a  distance  r  from  the  center  is  proportional  to  B^  —  r^.  What  is  the 
average  speed  over  a  cross  section  ?  What  is  the  quantity  of  flow  per 
unit  time  across  any  section  ? 

37.  The  kinetic  energy  E  ot  a,  moving  mass  is  lim  ]^Am  •  v^/2,  where 
Am  is  the  element  of  mass  moving  with  speed  v.  Show  that  for  a  disk  rotat- 
ing with  angular  speed  w,  E  =  w~I/2.  Calculate  E  for  a  solid  car  wheel 
of  steel,  30  in.  in  diameter  and  4  in.  thick  when  the  car  is  going  20  m./hr. 

38.  Show  that  the  kinetic  energy  J?  of  a  sphere  rotating  about  a  diame- 
ter with  angular  speed  w  is  (l/5)(mass)r2w2. 

39.  Calculate  the  kinetic  energy  in  foot-pounds  of  the  rim  of  a  flywheel 
whose  inner  diameter  is  3  ft,,  cross  section  a  square  6  in.  on  a  side,  if  its 
angular  speed  is  100  R.  P.  M.  and  its  density  is  7. 

40.  The  x-component  of  the  attraction  between  two  particles  m  and  m'. 
separated  by  a  distance  r,  is  (k  ■  m  ■  m'/r^)  cos  (r,  x)  where  cos  (r,  x)  de- 
notes the  cosine  of  the  angle  between  r  and  the  x-axis.  Hence  the  x-com- 
ponent of  the  attraction  between  two  elementary  parts  of  two  solids  M  and 
M'  is  (^-  •  AM-  AM'/f')  cos  (r,  x).  Show  thUt  the  total  attraction  between 
the  two  solids  is  expressible  by  a  six-fold  integral. 

41.  A  uniform  rod  attracts  an  external  particle  m.  Calculate  the  com- 
ponents of  the  attraction  parallel  and  perpendicular  to  the  rod  ;  the  re- 
sultant attraction  and  its  direction. 

[Hint.  Let  A3/ be  an  element  of  the  rod  ;  then  AF=  kAM  ■  m/r"^  is 
the  force  due  to  AM  acting  on  m,  r  being  the  distance  from  AM  to  m  ;  then 
the  components  of  AF  are  AX  =  AFcos  «  and  AY  —  Ai^'sin  a,  where  a 
is  the  angle  between  r  and  the  rod.     Hence 

X  =  J^cosa,  and  r  =  j*^sina.] 

42.  A  force  at  0  attracts  a  particle  at  P  proportionally  to  the  nth 
power  of  the  distance  OP.     What  \d  the  average  force  from  Pi  to  P2  ? 


CHAPTER   VIII 

METHODS    OF   APPROXIMATION 

PART   I.     EMPIRICAL   CURVES     INCREMENTS 

INTEGRATINCx  DEVICES 

119.  Empirical  Curves.  Some  of  the  methods  used  in 
science  to  draw  the  curves  which  represent  simultaneous 
values  of  two  related  quantities  and  to  obtain  an  equation 
which  represents  that  relation  approximately  are  given  in 
Analytic  Geometry.  Usually  the  pairs  of  corresponding  values 
are  plotted  on  squared  paper  first ;  in  all  that  follows  it  is 
assumed  that  this  has  been  done  in  each  case. 

120.  Polynomial  Approximations.  It  is  advantageous  to 
have  equations  which  are  as  simple  as  possible.  From  experi- 
mental results,  it  is  not  to  be  expected  that  absolutely  precise 
equations  can  be  found,  and  the  attempt  is  made  to  get  an  equa- 
tion of  simple  form  which  approximately  represents  the  facts,  in 
so  far  as  the  facts  themselves  are  known.  One  simple  kind  of 
function  which  often  does  approximately  express  the  facts  is  a 
polynomial : 

(1)  y  =  a-\-'bx  +  ca?  +  cW-{- h  kx". 

121.  Review  of  Elementary  Methods.  If  the  points  lie 
reasonably  close  to  some  straight  line,  it  is  usual  to  assume 
71  =  1  in  (1),  §  120,  whence  y  =  a-\-hx\  then  h  (the  slope)  and 
a  (the  ^/-intercept)  may  be  found  by  direct  measurements  in 
the  figure,  or  by  one  of  the  more  general  methods  which 
follow. 

227 


228  APPROXIMATION  [VIII,  §  121 

If  the  curve  has  the  typical  form  of  a  parabola,  it  is  advan- 
tageous to  assume  that  the  equation  is  of  the  form 

(2a)  y  =  a-\-hx  +  cx\     or     (26)   (ji  -  B)=  C(x  -  Af 

and  then  apply  the  methods  of  Analytic  Geometry  to  find  a,  b, 
c,  or  A,  B,  C.  One  of  the  methods  most  often  used  is  to  find 
a,  b,  c,  by  assuming  that  the  curve  actually  passes  through 
three  given  points  (see  Ex.  2,  p.  230). 

Another  method  that  can  be  be  used  whenever  the  vertex  of 
the  parabola  is  clearly  indicated,  is  based  on  the  fact  that 
(A,  B)  are  precisely  the  coordinates  of  the  vertex,  and  can 
therefore  be  measured  directly.  The  value  of  C,  which  is  all 
that  remains  to  be  found,  can  be  obtained  approximately  by  a 
variety  of  methods  :  one  may  lay  over  the  experimental  figure  a 
sheet  of  transparent  (tracing)  paper  on  which  the  curves 
?/=  kx^  have  been  drawn  for  a  large  number  of  values  of  Ti-.  or 
one  may  proceed  as  in  §  122 ;  or,  finally,  as  in  §  124,  below.* 

In  general,  the  equation  (1)  contains  ?i  + 1  unknown  coeffi- 
cients. To  obtain  these  values,  it  is  possible  to  use  any  n-\-l 
points  on  the  experimental  curve,  as  in  Analytic  Geometry. 
In  doing  so,  it  is  preferable  to  take,  not  the  precise  figures 
given  by  the  experiment,  but  rather  pairs  of  coordinates  of 
points  on  a  free-hand  curve  sketched  into  the  figure. 

General  formulas  for  the  values  of  the  coefficients  have  been  worked 
out,  and  are  given  in  the  Tables,  II,  I,  17,  under  the  name  Lagrange's 
Interpolation  Formula. 

In  the  theory  of  probabilities,  formulas  are  derived  (which 
are  to  be  found  in  any  large  set  of  mathematical  tables)  for  the 
most  probable  values  of  the  coefficients  a,  b,  c,  d,  etc.  These 
formulas  can  be  applied  by  any  person  even  before  studying 
the  theory.     See  Tables,  II,  D,  4. 

*  In  any  method,  judgment  on  the  part  of  the  exi^erimenter  is  the  final 
means  of  decidin^'whether  the  equation  obtained  will  approximately  repre- 
sent the  facts.  The  amount  of  error  which  may  exist  in  the  experimental 
measurements  is,  of  course,  fundamentally  important. 


VIII,  §  122]  EMPIRICAL   CURVES  229 

A  few  simple  problems  have  been  solved  already  by  one  of 
the  methods  of  probabilities :  in  Exs.  18-23,  p.  69,  we  assumed 
a  formula  of  the  type  y  =  kx,  and  found  A;  by  the  requirement 
that  the  sum  of  the  squares  of  the  errors  should  be  a  minimum. 
This  method  is  called  the  method  of  least  squares;  see  also 
Example  2,  §  105. 

122.  Logarithmic  Plotting.  The  preceding  forms  of  equa- 
tions may  not  represent  the  facts  very  well  unless  a  large  num- 
ber of  terms  of  (1),  §  120,  are  used. 

If  the  first  graph  resembles  one  of  the  curves  y  =  x-,  y  =  a^, 
y  =  X*,  etc.,  ov  y  =  o^'-,  y  =  o^'^,  etc.,  or  y  =  l/a*,  y  =  1/x^,  etc., 
it  is  advantageous  to  plot  the  common  logarithms  of  the 
quantities  measured  instead  of  the  actual  values  of  those 
quantities. 

If  X  and  y  represent  the  quantities  measured,  and  u  =  logioX, 
v  =  logioT/  are  their  common  logarithms,  the  values  of  u  and  v 
may  lie  very  nearly  on  a  straight  line, 

(1)  V  =  a-\-  bu, 

where  a  and  h  are  found  as  in  §  121.  Then  from  (1),  since 
u  =  logio  X,  V  =  logio  y, 

(2)  logio  y  =  a  +  h  logio  X  =  logio  ^'  +  logio  ^  =  logio  (^"a;^), 

where  logio  ^'  =  « ;  hence 

(3)  y  =  kx^. 

This  form  of  equation  is  very  convenient  for  computation  and 
is  used  in  practice  very  extensively  wherever  the  logarithmic 
graph  is  approximately  a  straight  line.*  This  work  applies 
equally  well  for  negative  and  fractional  values  of  h. 

*  To  avoid  the  trouble  of  looking  up  the  logarithms,  a  special  paper  usually 
described  in  Analytic  Geometry  may  be  purchased  which  is  ruled  with  loga- 
rithmic intervals.  No  particular  explanation  of  this  paper  is  necessary  e.Kcept 
to  say  that  it  is  so  made  tliat  if  the  vahies  of  x  and  ?/  are  plotted  directly,  the 
graph  is  identical  with  that  described   above.    To  secure   this  result   the 


230  APPROXIMATION  [VIII,  §  123 

In  many  cases  where  the  process  just  described  fails,  it  is  sometimes 
advantageous  to  assume  that  the  equation  has  the  form  (y—B)  =  k{x—A)" 
which  evidently  has  a  horizontal  tangent  at  the  point  (A,  B)  ii  n>l,  or 
a  vertical  tangent  if  n  <  1.  If  the  first  graph  (in  x  and  y)  shows  such  a 
vertical  or  horizontal  tangent,  that  point  {A,  B)  may  be  selected  as  a  new 
origin,  and  the  values  x'  =  x  —  A  and  y'  —  x—  B  should  be  used ;  thus 
we  would  plot  the  values  of 

u  =  logio  x'  =  logio  (x-A),     v  =  logio  y'  =  logio  (.y  -  B), 
in  the  manner  described  above.     The  values  of  A  and  B  are  found  from 
the  first  graph  (in  x  and  y) ;  the  values  of  k  and  n  are  found  from  the  log- 
arithmic graph  as  above, 

123.  Semi-logarithmic  Plotting.  Variations  of  this  process 
of  §  122  are  described  in  Exercises  XLIX  below.  In  par- 
ticular, if  the  quantities  are  supposed  to  follow  a  compound  in- 
terest law,  y  =  fce*"^,  it  is  advantageous  to  take  logarithms  of  both 
sides :  log^^  y  ^  log^^  k  +  bx  log^  e, 

and  then  plot  m  =  x,  v  =  logm?/ ;  if  the  facts  are  approximately- 
represented  by  any  compound  interest  law,  the  experimental 
graph  (in  u  and  v)  should  coincide  (approximately)  with  the 
straight  line  v  =  A-\- Bu, 

where  A  =  logio  ^  and  5  =  6  logjo  e.  After  A  and  B  have  been 
measured,  k  and  b[=B  log^  10  =  2.303  B]  can  be  found. 

EXERCISES  XLIX. —EMPIRICAL   CURVES:    ELEMENTARY  METHODS 

1.  Find  the  equation  of  a  straight  line  through  the  points  (—1,  3) 
and  (2,  5);  through  (2,  -  3)  and  (4,  5). 

2.  Determine  a  parabola  whose  axis  is  vertical,  through  the  three 
points  (0,3),  (2,-1),  (5,  8). 

[Hint  :  Assume  the  equation  in  each  of  the  forms  y  =  ax'^  +  bx  +  c, 
y  —  B  =  C  (x  —  A)'^ ;  check  the  answers  by  comparing  them.] 

successive  rulings  are  drawn  at  distances  proportional  to  logl(=^0),  log  2, 
log  3,  ••    from  one  corner,  both  horizontally  and  vertically. 

Explanations  and  numerous  figures  are  to  be  found  in  many  books ;  see, 
e.g.,  Kent,  "Mechanical  Engineers'  Pocket  Book "  (Wiley,  1910) ,  p.  8.5 ;  Traut- 
wine,  "Civil  Engineers'  Pocket  Book"  (Wiley),  (Chapter  on  Hydraulics). 


VIII,  §  123]  EMPIRICAL  CURVES  231 

3.  Determine  a  cubic  function  of  x  which  takes  on  the  values  —  10, 
—  2,  6,  20,  respectively,  when  a;  =  —  1,  0,  1,2. 

4.  Determine  n  and  c  so  that  the  curve  y  =  ex"  passes  through  the  two 
points  (1,  2)  and  (3,  54);  through  (1,  3),  (4,  6);  through  (1,  3),  (8,  12). 

5.  Plot  the  data  of  Ex.  18,  List  XIV,  p.  69  ;  draw  a  straight  line  as 
closely  as  possible  through  the  points  without  giving  a  preference  to  any 
one  ;  determine  the  equation  from  this  graph  ;  compare  it  with  the  result 
obtained  in  List  XIV. 

6.  Proceed  as  in  Ex.  5  for  each  of  the  cases  in  Exs.  19-23,  List  XIV. 

7.  Assuming  the  data  of  Ex.  1,  §  124,  p.  235,  find  graphically  the 
equation  connecting  /  and  w  and  compare  it  with  the  result  found  in  §  124. 

8.  Assuming  the  data  of  Ex.  2,  §  124,  sketch  a  parabola  whose  axis  is 
parallel  to  the  axis  of  0  ;  determine  its  equation  ;  compare  the  result  with 
that  of  §  124. 

9.  Find  a  parabolic  curve  of  the  second  degree  which  coincides  vrith 
y  —  sin  X  at  the  points  where  a;  =  0,  x  =:  ir/2,  x  =  ir.  Compare  the  areas 
under  the  two  curves. 

10.  Proceed  as  in  Ex.  9  for  each  of  the  following  curves,  taking  the 
values  of  x  specified  in  each  case  : 

(a)  y  =  logio  X,  X  =  1,  x  =  5,  x  =  10. 

(b)y  =  &',  x=— 1,  x  =  0,  x  =  +l, 

(c)  y  =  tan  x,  x  =  0,  x  =  ir/S,  x  =  ir/G. 

{d)  y  -x^  —  7  x  +  2,  x-0,  x  =  2,  x  =  4. 

11.  Find  a  parabolic  curve  of  the  third  degree  through  four  points 
taken  at  equal  horizontal  intervals  on  the  curve  y  =  sin  x,  between  x  =  0 
and  X  =  ir/2.    Compare  the  areas  under  the  two  curves. 

12.  Find  a  parabolic  curve  of  the  second  degree  which  coincides  with 
y  —  sin  X  at  X  =  0  and  x  =  7r/2,  and  which  has  the  same  slope  as  y  =  sin  x 
at  X  =  0. 

13.  Find  a  polynomial  of  second  degree  which,  together  with  its  first 
and  second  derivative,  coincides  with  cos  x  at  x  =  0. 

14.  Proceed  as  in  Ex.  12  for  the  curve  y  =  e'. 

15.  Find  a  cubic  which,  together  with  its  first  three  derivatives,  coin- 
cides with  each  of  the  following  functions  when  x  =  0  : 

(a)  sinx,  (6)  tanx,  (c)  e*,  C^^)  1/(1  +  x). 


232 


APPROXIMATION 


[VIII,  §  123 


16.  Plot  each  of  the  following  curves  logarithmically,  —  either  by  plot- 
ting logio  X  and  logio  ?/,  or  else  by  using  logarithmic  paper : 

(a)  y  =  2  x^.  (c)  y  =  A  x^-^.  (e)    y  =  5.7  x«. 

(6)  2/ =3x1/2.  (d)y  =  3x-^.  (/)  2/ =  - 1.4x2-4. 

17.  In  each  of  the  following  tables,  the  quantities  are  the  results  of 
actual  experiments;  the  two  variables  are  supposed  theoretically  to  be 
connected  by  an  equation  of  the  form  y  =  kx":  Draw  a  logarithmic  graph 
and  determine  k  and  n,  approximately : 

(a)  [Steam  pressure  ;  w  =  volume,  p  =  pressure.]    [Saxelby]. 


V 

2         4 

6 

8 

10 

p 

68.7       31.3 

19.8 

14.3 

11.3 

(6)  [Gas  engine  mixture  ;  notation  as  above.]     [Gibson.] 

3.54        4.13       4.73       5.35       5.94       6.55       7.14      7.73      8.04 


141. 


115 


95 


81.4       71.2 


54.( 


50.7       45 


(c)  [Head  of  water  ft,  and  time  t  of  discharge  of  a  given  amount.] 
[Gibson.] 


h 

0.043 

0.057 

0.077 

0.095 

0.100 

t 

1260 

540 

275 

170 

138 

(d)  [Heat  conduction,  asbestos;  ^  =  temperature  (F.),  C  =  coefficient 
of  conductivity.]     [Kent.] 


e 

32° 

212° 

392° 

572° 

7.52°     1112° 

c 

1.048 

1.346 

1.451 

1.499 

1.548     1.644 

VIII,  §  124]  EMPIRICAL  CURVES  233 

(e)  [Track  records  :  d  =  distance,  t  =  record  time  (intercollegiate).] 


d 

100  yd. 

220  yd. 

440  yd. 

880  yd. 

Imi. 

2  mi. 

t 

0:09f 

0:211 

0:48| 

l;56 

4:17i 

9:27| 

[Note.     See  KenneUy,   Fatigue,   etc.,  Proc.  Amer.  Acad.  Sc.  XLII, 
No.  15,  Dec.  1906  ;  and  Popular  Science  Monthly,  Nov.  1908.] 

18.  Plot  the  following  curves,  using  logarithmic  values  of  one  quantity 
and  natural  values  of  the  other  : 


(«)  y 


(6)  y  =  10e3«. 


19.  Discover 
sets  of  data : 

a  formula  of  the  type 

y  =  ^•e<" 

for  each  of  the  fol 

(a) 

{;: 

.2 
1.6 

.4 

2.2 

.6 
3.3 

.8 
6.0 

1.0 
7.4 

if>) 

'  X: 

y- 

.6 
3.0 

1.2 
4.4 

1.8 
6.6 

2.4 
10.0 

3.0 
14.8 

(c) 

\7: 

.31 
1.22 

.63 
1.49 

.94 
1.82 

1.26 
2.23 

1.57 
2.72 

((0 

X: 

.2 

.82 

.8 
.45 

2.0 
.13 

4.0 
.02. 

(e) 

X  : 

y- 

.63 
2.01 

1.26 
1.35 

2.51 
.60 

3.77 

.27 

5.03 
.12 

(/) 

'  x: 

y- 

1 
1.63 

2 
1.34 

3 
1.08 

4 

.90 

5 
.73 

20.  A  is  the  amplitude  of  vibration  of  a  long  pendulum,  t  is  the  time 
.since  it  was  set  swinging.  Show  that  they  are  connected  by  a  law  of  the 
form  A  =  ke-~^. 

Ain.=       10  4.97  2.47  1.22  .61  .30  .14 

fmin.=:         0  12  3  4  5  6 

124.    Method  of  Increments.     A  method  which  is  often  better 
in  practice  than  those  in  §  121  is  as  follows.     If  the  curve  is 
supposed  to  be  a  parabola, 
(1)  y  =  a  +  bx  +  CX-, 


234  APPROXIMATION  [VIII,  §  124 

and  if  we  take  two  pairs  of  values  of  x  and  y,  say  (x,  y)  and 
{x  +  dkX,  y  +  A^/)  given  by  experiment,  we  should  have 

(2)  y  =  a  +  hx-{-  ex-,     y  +  d^y  =  a-\-h{x  +  ^x)  -\-  c{x-\-  Axy, 

whence 

(3)  Ay  =  b  Aa;  +  2  ex  Ax  +  c  Ace". 

If  Ao;  is  constant,  i.e.  if  points  are  selected  at  equal  intervals 
on  the  crudely  sketched  curve  drawn  through  the  experimental 
points,  we  might  write 

(4)  Y=Ay=(bh-\-ch^)-\-2ch-x  =  A  +  Bx 

where  h  =  Aa;.  If  we  should  actually  plot  this  equation, 
Y=A-\-Bx,  we  would  get  (approximately)  a  straight  line. 
Now  Ay  =  Yis  the  difference  of  two  values  oi  y;  it  can  be 
found  for  each  of  the  values  of  x  selected  above,  and  the 
(approximate)  straight  line  can  be  drawn,  so  that  A  and  B 
can  be  measured  as  in  §  121. 

We  may  repeat  the  preceding  process ;  from  (4)  we  obtain, 
as  above, 

(5)  AY=BAx  =  2  ch%     (h  =  Ax), 

whence  AF  is  constant  if  h  was  taken  constant.  Now  AF  is 
the  difference  between  two  values  of  Y;  that  is,  AF  is  the 
difference  between  two  values  of  Ay : 

AY=A{Ay)  =  A% 

and  for  that  reason  is  called  a  second  difference,  or  a  second 
increment.  If  the  second  differences  are  reasonably  constant, 
we  conclude  that  an  equation  of  the  form  (1)  will  reason- 
ably represent  the  facts  and  we  find  c  directly  by  solving 
equation  (5). 


VIII,  §  124] 


EMPIRICAL  CURVES 


235 


Example  1 .     With  a  certain  crane  it  is  found  that  the  forces  /  measured 
in  pounds  which  will  just  overcome  a  weight  w  are 


/ 

8.5 

12.8 

17.0 

21.4 

25.6 

29.9 

34.2 

38.5 

w 

100 

200 

300 

400 

500 

600 

700 

800 

What  is  the  law  connecting  power  with  the  weight  that  it  just  overcomes  ? 

[Pkrry.] 
Plotting  the  values  of  /and  ip,  it  appears  that  the  points  are  very  nearly 
on  a  straight  line  f  —  a  -\-  bio.  If  they  were  on  a  straight  line,  Af/Aw 
would  be  constant  and  equal  to  df/dw  =  b.  As  a  matter  of  fact,  for  each 
increase  of  weight,  Af/Aw  varies  only  from  .042  to  .044,  its  average  value 
being  30/700  =  .0429.  Taking  this  value  for  b,  one  gets  for  the  equation 
of  the  line,  and  hence  for  the  relation  between  power  and  weight : 

/=  4.21  +  .0429  10,     4.21  =  8.5  -  100  x  .0429. 

Here  4.21  appears  to  be  the  power  needed  to  start  the  crane  if  no  load 
were  to  be  lifted. 

Example  2.     If  9  is  the  melting  point  (Centigrade)  of  an  alloy  of  lead 
and  zinc  containing  x  %  of  lead,  it  is  found  that 


X  =  %  lead 

40 

50 

60 

70 

80 

90 

6  =  melting  point 

186 

205 

226 

250 

276 

304 

Plotting  the  points  (a;,  d)  will  show  them  not  to  lie  in  a  straight  line  as 
ds  also  shown  by  the  differences  A^.  But  A  (A^)  or  A-^  does  run  uni- 
formly.    Therefore  one  tries  a  quadratic  function  of  x  for  6,  that  is 


It  is  evident  that 
and 


^  =  ffl  +  6x  +  cx^. 
=  10?^-f  c(20a;  +  100), 
A:^e  -  200  c. 


The  average  value  of  A2fl  is  2.25.  Hence  c  =  .01125.  If  we  subtract 
ca;2  from  6,  we  find  6  —  cx!^  =  a  +  bx.  These  values  can  be  calculated  from 
the  data  and  from  c  =  .01125  ;  they  will  be  found  to  lie  on  a  straight  line  ; 


2.5 

3.0 

2.1 

2.8. 

.12 

.14 

.136 

.163 

236  APPROXIMATION  [VIII,  §  124 

hence  a  and  b  can  be  found  by  any  one  of  several  preceding  methods. 
The  student  will  readily  obtain,  approximately, 

e=  133 +  . 875  x  +  . 01125x2, 

a  formula  which  represents  reasonably  the  melting  point  of  any  zinc-lead 
alloy.  [Saxelby.] 

EXERCISES  L.  — EMPIRICAL   CURVES  BY  INCREMENTS 

1.  Express /(x)  as  a  quadratic  function  of  x,  when 
x:  0  0.5  1.0  1.5  2.0 

/(x):  2.5  1.9  1.6  1.5  1.7 

2.  Express /(x)  as  a  cubic  function  of  x,  when 
x:  0         .02  .04  .06  .08  .10 

/(x):  0         .020         .042         .064         .087         .111 

3.  Express  ^Cw)  as  a  cubic  in  tji,  when 

to:     .01  .02  .03  .04  .05         .06  .07  .08 

<{>{m):     .00010     .00041     .00093     .00166     .00260  .00385     .00530     .00690. 

4.  The  specific  heat  8  of  water,  at  6°  C,  is 

6:        0  5  10  15  20  25  30 

S:  1.0066       1.0038       1.0015       1.0000      0.9995       1.0000       1.002. 
Express  S  in  terms  of  d. 

5.  Determine  a  relation  between  the  vapor  pressure  P  of  mercury, 
and  the  temperature  5  C,  from  the  data  below  : 

^:  60  90  120  150  180  210  240 

P:  .03  .16  .78  2.93  9.23  25.12  58.8. 

6.  The  resistance  R^  in  ohms  per  1000  feet,  of  copper  wire  of  diame- 
ter D  mils,  is  * 

D:  289  182  102  57  32  18  10 

B:  .126  .317  1.010  3.234  10.26  32.8  105.1. 

Find  a  relation  between  B  and  D. 

7.  The  Brown  and  Sharpe  gauge  numbers  N  of  wire  of  diameter  D 
mils,  are 

iV:  1  5  10  15  20  25  30 

D:  289  182  102  67  32  18  10. 

Express  D  in  terms  of  N. 


VIII,  §  124]  EMPIRICAL  CURVES  237 

8.  Find  a  relation  between  tlie  speed  <S'  of  a  train  in  kilometers  per 
hour,  and  the  horse-power  (H.  P.)  of  the  engine  from  the  data  below  : 

H.  P.  :  550  650  750  850 

S:  2Q  35  52  70. 

9.  Determine  a  relation  between  the  age  of  a  lamp  and  its  candle 
power  (C.  P.)  from  the  following  data: 

Hours:    0  250  500         750  1000        1250        1500 

C.P.  :24.0         17.6         16.5         15.8         15.3         14.9         14.5. 

10.  Proceed  as  in  Ex.  9  for  each  of  the  two  lamps  (I  and  II)  below  : 
Hours:  0  250  500  750  1250  1750  2250  2750 
C.P.       I:  13.70       15.80     16.65     16.50     14.50       13.25       12.00       11.40 

H:  17.75       20.00     19.00     18.60     17.90       17.00       15.50       14.10. 

11.  The  dip,  0,  of  the  magnetic  needle  at  Harrisburg,  Pa.,  was  ob- 
served as  below : 

Date:  1840.5  1862.6  1877.7  1885.6  1895.7 

Dip:  72°.34  72''.50  72°.34  71°.75  71°. 72. 

Show  that  e  =  72°.48  +°  .0067  m  -  °.00056  m^,  where  m  =  date  -  1850. 
At  what  date  was  the  dip  greatest  ? 

12.  Proceed  as  in  Ex.  11  for  the  data  below,  taken  at  Eastport,  Me.: 
Date:  1860.5       1863.5       1865.6       1873.7        1879.6       1887.6       1895.6 

Dip:75°.88        75^.80        75°.74        75°.41        75°.20       74°.90        74=^.63. 
Show  that  d  =  76°.31-0°.039  m  +°  .000053  m%  where  m  =  date  —  1850. 

13.  The  intensity  of  illumination  at  the  same  distance  but  in  different 
directions  from  an  incandescent  lamp  was  observed  as  below,  6  =  0'^ 
being  downward  and  0  —  90°  horizontally  from  the  lamp  : 

0:0  30  60  90  120  150 

C.  P.  :  6.6  9.5  14.5  16.0  14.5  9.8. 

Lay  off  C.  P.  and  0  in  rectangular  and  also  in  polar  coordinates,  and  find 
a  relation  between  them. 

14.  Proceed  as  in  Ex.  13  for  a  shaded  lamp  : 

^ :     0  10  20  30  40  50  60  70 

C.  P.  :  47.3         44.2         36.6  30.0         25.6         22.0         17.6         10.6. 


238 


APPROXIMATION 


[VIII,  §  124 


15.  The  energy  consumed  in  overcoming  molecular  friction  when  iron 
is  magnetized  and  demagnetized  (hysteresis,  H,  —  measured  in  watts  per 
cycle  per  liter  of  iron)  is  given  below  in  terms  of  the  strength  of  the 
magnetic  field  {B,  —  measured  in  lines  per  square  centimeter).  What  is 
the  relation  between  them  ? 


B:  2000 
H:  .022 


4000 
.048 


6000 


8000 
.138 


10000 
.185 


14000 
.320 


16000        18000 
.400  .475. 


16.  Proceed  as  in  Ex.  15,  for  cobalt,  the  hysteresis  loss  H  being  now 
measured  in  ergs  per  cycle  per  second  : 

B:  900         2350         3100         4100         4600        5200  5850  6500 

H:  450         2450         3950         6300         7400        8950         10950         13250. 

17.  The  table  below  contains  some  data  on  the  comparison  of  a  tung- 
sten lamp  with  a  tantalum  lamp.  The  voltage  or  electrical  pressure  F,  is  in 
volts,  the  resistance  iJ,  in  ohms,  the  current  consumed  in  watts  per  candle 
power  ;  C  denotes  candle  power,  and  W  watts  per  candle  power. 


Tungsten 

Tantalum 

Voltage 

C.  P. 

Watts 
per  C.  P. 

Kesi  stance 

C.  P. 

"Watts 
per  C.  P. 

Resistance 

V 

C 

W 

R 

C 

W 

B 

80 

14 

2.51 

166 

5 

3.80 

260 

90 

24 

1.83 

173 

10 

2.85 

265 

100 

36 

1.49 

182 

18 

2.05 

275 

110 

52 

1.23 

190 

25 

1.65 

283 

120 

71 

1.10 

197 

38 

1.35 

290 

130 

95 

0.96 

202 

50 

1.15 

300 

140 

128 

0.83 

210 

62 

0.95 

308 

150 

160 

0.76 

216 

78 

0.85 

315 

160 

196 

0.58 

222 

100 

0.75 

323 

170 

230 

0.52 

227 

122 

0.70 

327 

180 

270 

0.50 

232 

156 

0.70 

332 

190 

312 

0.48 

238 

190 

0.60 

340 

200 

340 

0.47 

242 

235 

0.55 

345 

For  each  lamp,  express  each  of  the  quantities  C,  TF,  i?,  in  terms  of  V. 


VIII,  §  125]      APPROXIMATE  INTEGRATION 


239 


125.  Approximate  Integration.  One  method  of  finding  ap- 
proximate values  of  a  definite  integral  is  that  used  in  defining 
an  integral,  §  66,  p.  114.  This  consists  in  finding  special 
values  of  the  sums  S  and  s  of  p.  114,  by  breaking  up  the  inter- 
val between  the  limits  into  many  parts,  and  combining  portions 
approximately  as  if  they  were  rectangles : 


(1) 


S  =  :S.f(xj,)  Ax,    s  =  2/(a-^)  Ax, 


where  x^  and  x^  denote,  respectively,  the  values  of  x  at  the 

right  and  left  ends  of  the  partial 

interval. 

A  still  better  value  is  obtained 
by  averaging  these  two : 


(2) 


S  + 


.Srfi-rn)+f(Xr)^^^ 


for   this    amounts    to    the    same 
thing  as  replacing   each   partial 

area  by  the  trapezoid  whose  base  is  Ax  and  whose  sides  are 
f(xji)  a.udf(x^).     See  footnote,  p.  112, 
Finally  the  prismoid  rule  (§  71,  and  Ex.  10,  p.  129)  gives 


(3)         j      /(.«)  dx  = ^-^ (&  -  a), 


which  amounts  to  replacing  the  curve  by  a  parabolic  arc 
y  =  A3?  +  Bx  -\-  C  through  its  end  points  (x  =  a  and  x  =  h) 
and  its  middle  point  x  =  (a-f-  6)/2. 

If  the  prismoid  rule  is  applied  to  each  successive  pair  of 
an  even  number  of   subdivisions   of   width    Ax  each,  and   if 


X  —  .Tj,     X^)     Xq, 


x„_i  be   the  values   of  x  at  the   division 


240  APPROXIMATION  [VIII,  §  125 

points,  we  find,  approximately, 
(4)    r'~'f{x)dx 

%Jx=a 

which  is  known  as  Simpson's  Rule.     See  Ex.  12,  p.  129. 

All  these  rules  evidently  apply  to  the  approximate  computa- 
tion of  any  integral,  no  matter  where  it  arose. 

126.  Integration  from  Empirical  Formulas.    Limit  of  Error. 

If  a  formula  y  =f(x)  has  been  obtained  empirically,  it  may  be 
used  to  find  the  area  under  the 
curve  represented  by  the  experi- 
mental data.  If  the  maximum 
M  error  due  to  experimental  errors 
and  to  faulty  approximation  is 
Fig.  53  ^  ^^  ^^^^  ^Yiq  true  value  of  y 

differs  from  f{x)  by  at  most  M,  we  have,*  if  6  >  a, 

I       ydx\<\\      f{x)dx\  +  \  I       Mdx\ 

\Jx-a  \  I  «-'i=a  I  I  \Jx=a  \ 

^\£_^J{x)dx\+Mi^-ay, 

that  is,  the  error  in  the  value  of  the  integral  calculated  by 
using  the  approximation  formula  y  =f(x)  is  not  greater  than 
M(b  -  a). 

*  The  pair  of  vertical  lines  |  I  indicate,  as  before  (see  pp.  16,  171),  the  posi- 
tive numerical  value  (or  absolute  value)  of  the  quantity  inclosed. 


VIII,  §  127]      APPROXBIATE   INTEGRATION  241 

The  same  result  applies  in  cases  in  which  a  function  to  be 
Integrated  has  been  replaced,  for  convenience,  by  a  simpler 
function. 

Thus  _J_z=l +  x  +  a;2  + -^. 

1  -  X  1  —  X 

If  we  replace  1/(1  —  x),  for  convenience,  by  1  +  a;  +  x'^,  the  error  E 
made  in  doing  so  is  : 

l-x 

which,  for  values  of  x  numerically  less  than  1/10,  is  numerically  less  than 
(.l)V->  <  -0012  ;  hence  if  we  write 

the  en-or  £made  in  the  value  of  the  integral  is  less  than  .1  ■  .0012  =  .00012. 
The  exact  value  of  the  original  integral  is 

-  log  (1  -  x)J^=^  -  log  (.9)  =  -  logio  (.9)  loge  10  =  .045757  •  2.30258  = 

.10536.     In  general,  as  in  the  example,  the  final  error  may  be  very  much 
le.ss  than  the  estimated  upper  limit  of  the  error  calculated  above. 

127.  Derived  and  Integral  Curves.  In  §  49,  p.  77,  we  drew 
the  derived  curves  by  finding  the  derivatives  and  plotting 
their  values. 

If  the  original  curve  was  drawn  from  values  found  by  some  experi- 
ment, and  if  its  equation  is  unknown,  the  derived  curve  can  be  drawn 
mechanically.  To  do  so,  draw,  according  to  your  be.st  judgment,  the 
tangents  at  each  of  a  large  number  of  points  (xo,  yo),  (Xit  Vi),  (xo,  1/2), 
•••»  («ni  Vn),  noting  about  how  much  uncertainty  there  seems  to  be  in 
each  case.  Find  the  slope  m,-  of  the  tangent  at  each  point  (x,-,  ?/,■)  by 
measuring  its  rise  per  horizontal  unit.  Plot  the  points  (m,-,  x,),  indicat- 
ing the  estimated  uncertainty  in  each  value  of  m.  Draw  a  smooth  curve 
which  passes  near  each  of  these  points,  allowing  the  most  variation  at 
the  points  where  the  values  of  ??i  seemed  to  be  most  uncertain.  Check 
by  comparing  the  slope  of  the  original  curve  and  the  ordinate  of  the 
derived  curve  for  various  other  values  of  x.  This  process  may  not  be 
very  reliable,  and  every  possible  check  must  be  used.     (See  §  143(d).) 


242  APPROXIMATION  [VIII,  §  127 

Likewise,  if  any  function  y  =f(x)  is  given,  the  integral 
curve :  . 

I  =  jf(x)clx  =  <f>(x)  +  0, 

which,  represents  the  area  under  y  =f(x)  from  some  fixed  left- 
hand  boundary  to  the  ordinate  x  =  x  can  be  drawn.*  But  if 
the  equation  of  the  curve  is  not  known,  this  can  still  be  done 
by  the  methods  of  §  125 ;  or  by  simply  estimating  the  area 
from  some  left-hand  vertical  line  up  to  various  points  a^,  iCj, 
•  ••,  x„  and  marking  at  each  value  of  x,  as  a  new  ordinate,  the 
value  of  the  area  up  to  that  point.  The  result  is  surprisingly 
accurate  if  the  curve  is  drawn  on  millimeter  paper  and  the 
area  obtained  by  actually  counting  the  squares.  The  accuracy 
of  this  process  as  compared  with  the  uncertainty  of  mechanical 
construction  of  the  derived  curves,  is  a  consequence  of  §  126. 

EXERCISES  LI.— APPROXIMATE  EVALUATION  OF  INTEGRALS 

1.  rind  the  area  iinder  the  curve  y  =  1/(1  —  x)  from  a-  =  0  to  a:  =  .1 
by  use  of  the  prismoid  formula,  and  show  that  the  result  is  accurate  to 
five  decimal  places. 

2.  Draw  the  curve  y=  1/(1  —x)  and  construct  the  integral  curve 
from  X  =  0  to  any  value  of  x  less  than  1,  first  by  actually  counting  the 
squares  on  the  cross  section  paper,  second  by  actually  integrating  betvi^een 
the  limits  x  =  0  and  x  =  x. 

3.  rind  the  area  under  the  curve  ?/  =  l/x^  between  x  =  l  and  x  =  2, 
approximately,  first  by  using  the  prismoid  formula,  then  by  using  Simp- 
son's rule  with  three  intermediate  points  of  division.  Compare  the  results 
with  the  precise  answer  obtained  by  integration,  ^ 

4.  Find  the  error  made  in  computing  the  value  of  the  area  of  one 
arch  of  the  curve  y  =  sin  x  if  the  approximating  parabola  of  Ex.  9,  List 
XLIX,  p.  231,  is  used  instead  of  the  sine  curve. 

5.  Proceed  as  in  Ex.  4,  for  each  of  the  curves  and  their  approximating 
parabolas  mentioned  in  Ex.  10,  List  XLIX,  taking  the  extreme  values  of  x 
mentioned  there  as  limits  of  integration. 

*  Different  values  of  C  give,  of  course,  different  integral  curves,  all  con- 
gruent, obtained  from  any  one  of  them  by  a  stiff  vertical  motion. 


VIII,  §  12S]      APPROXIMATE  INTEGRATION  243 

6.  Show  that  a;2.5  ijes  between  x-  and  x'^  from  x  =  0  to  .r  =  1.  Hence 
show  that  f  a;2.5  c^^  lies  between  1/3  and  1/4.  Find  the  exact  vahie  of 
the  integral. 


7.   Show  that  1/Vl  -  x*  lies  between  1/Vl-x-  and  l/\/2(l  -  x^) 
between  x  =  0  and  x  =  1  ;    hence  find  extreme  limits  between  which 


£ 


[l/Vl-x*]dxlies. 

8.  Compute  the  value  of  the  integral    T  [1/(1  +  x2)](Zx    (a)   by  the 

prismoid  rule  ;  (b)  by  the  trapezoid  rule,  with  two  intermediate  points  of 
division  ;  (c)  by  Simpson's  rule,  with  three  intermediate  points  of  division ; 
(rf)  precisely  by  direct  integration.     Compare  the  results  for  accuracy. 

9.  Show  by  long  division  that  1/(1  +  ,r-)  =  1  —  x~  +  x*  -  x^/(l  +  X'^). 
Hence  show  that  the  area  under  y  —  1/(1  +  x'-)  from  x  =  0  to  x  =  .5  dif- 
fers from  that  under  y  =  1  —  x^  +  x*  by  less  than  1/128.  Actually  com- 
pute both  areas,  and  show  that  this  estimate  of  the  error  is  far  larger  than 
the  actual  error. 

10.  Draw  the  curve  y  =  1/(1  +  x'-)  and  construct  its  integral  curve, 
starting  from  the  initial  point  x  =  0.     Verify  by  direct  integration. 

11.  Draw  the  curve  y  =  e'~^  and  construct  its  integral  curve.  Find  the 
value  of  the  integral  from  x=0  to  x  =  l,  approximately,  [a)  from  this  inte- 
gral curve  ;  (6)  by  the  prismoid  formula  ;  (c)  by  the  trapezoid  rule,  with 
one  intermediate  point ;  (d)  by  Simpson's  rule,  with  one  intermediate  point. 

12.  In  each  of  the  exercises  of  Ex.  17,  List  XLIX,  p.  232,  estimate  from 
the  figure  an  upper  limit  of  the  difference  between  the  given  data  and  the 
values  represented  by  the  empirical  formula  obtained.  Hence  find  an 
upper  limit  of  the  total  error  which  would  be  made  in  using  the  empirical 
formula  to  find  the  area  underneath  the  curve. 

13.  Given  a  function /(x)  defined  by  the  following  set  of  data  : 

X  0       .1       .2       .3       .4      *    .5  .(5  .7  .8  .9  1 

/(x)        1       .9      .7       .4        0      -.4       -.7      -.9      -1.       -.0      -1 

find  approximately  the  derivative  of  /(x)  at  each  of  the  points  x  =  .2, 

a;  =  .3,  x  =  .7.     Find   approximately  the  value  of  the  integral  of  /(x) 

from  X  =  0  to  each  of  the  preceding  values  of  x. 

128.  Integrating  Devices.  It  is  important  in  many  prac- 
tical cases  to  know  approximately  the  areas  of  given  closed 
curves.  Thus  the  volume  of  a  ship  is  found  by  finding  the 
areas  of  cross  sections  at  small  intervals. 


244 


APPROXIMATION 


[VIII,  §  128 


Besides  the  methods  described  above,  the  following  devices 
are  employed : 

A.  Counting  squares  on  cross  section  paper. 

B.  Weighing  the  figures  cut  from  a  heavy  cardboard  of 
uniform  known  weight  per  square  inch. 

C.  Integraphs.  These  are  machines  which  draw  the  inte- 
gral curve  mechanically;  from  it  values  of  the  area  may  be 
read  off  as  heights. 

The  simplest  such  machine  is  that  invented  by  Abdank-Abakanowicz. 
A  heavy  carriage   CDEF  on  large  rough  rollers,  R,  B'  is  placed  on  the 

paper  so  that  CE  is  par- 

iinii 


fM 


allel  to  the  y-axis. 

Two  sliders  S  and  S' 
move  on  the  parallel 
sides  DF  and  CE ;  to  S 
is  attached  a  pointer  P 
which  follows  the  curve 
y  =  f(x) .  A  grooved  rod 
AB  slides  over  a  pivot 
at  A,  which  lies  on  the 
.r-axis,  and  is  fastened  by 
pivot  B  to  the  slider  S. 
A  parallelogram  mechan- 
ism forces  a  sharp  wheel 
IV  attached  to  the  slider 
S'  to  remain  parallel  to 
AB.  A  marker  Q  draws 
a  new  curve  i  =  (p{x), 
which  obviously  has  a  tangent  parallel  to  W,  that  is,  to  AB.  If  AB  makes 
an  angle  a  with  Ox,  tan  «  is  the  slope  of  the  new  curve  ;  but  tan  «  is  the 
height  of  S  divided  by  the  fixed  horizontal  distance  h  between  A  and  B  : 

hei^htoTS^fJx). 
h  h    ' 


whence 


-'"=li7j'^''^^''' 


where  a  is  the  value  of  x  at  P  when   the  macbine  starts,  and  /„  denotes 
the  vertical  height  of  the  new  curve  at  the  corresponding  point. 


VIII,  §  128]      APPROXBIATE   INTEGRATION 


245 


D.  Polar  Planimeters.  —  There  are  machines  which  read  off 
the  area  directly  (for  any  smooth  closed  curve  of  simple  shape) 
on  a  dial  attached  to  a  rollini?  wheel. 


The  simplest  such  machine  is  that  invented  by  Amsler. 
Let  us  first  suppose  that  a  moving  rod  ab  of  length  I  always  remains  per- 
pendicular to  the  path  described  by  its  center  C.  The  path  of  C  may  be- 
regarded  as  the  limit  of  an  inscribed  polygon,  and  the  area  swept  over  by 
the  rod  may  be  thought  of  as  the  limit  of  the  sum  of  small  quadrilaterals, 
the  area  A^  of  each  of  which  is  lAp,  approximately,  where  Ap  is  the  length 
of  the  corresponding  side 
of  the  polygon  inscribed  in 
the  path  of  C.  Hence  the 
total  area  A  swept  over 
by  the  rod  is  evidently  Ip^ 
where  p  is  the  total  length 
of  the  path  of  C. 

But  if  the  rod  does  not 
remain  perpendicular  to  the 
path  of  C  during  the  motion,  and  if  ^  is  the  angle  between  the  rod  and 
that  path,  the  area  AA  becomes  Z  sin  <//  •  Ap,  approximately.  The  expres- 
sion sin  ^  •  Ajj  may  be  thought  of  as  the  compo- 
nent of  Ajj  in  a  direction  perpendicular  to  the  rod. 
Calling  this  component  As,  we  have  AA  =  ZAs, 
approximately  ;  and  the  total  area  A  swept 
over  by  the  rod  is  precisely  lira  SA^l  =  lim  'Zl\s 
=  j  Ids  =1  ^(ls  =  Is,  where  s  =  ^  ds  is  the  total 
motion  of  C  in  a  direction  perpendicular  to 
the  rod. 

The  quantity  s  =  i  ds  can  be  measured  me- 
chanically by  means  of  a  wheel  of  which  the  rod 
is  the  axle,  attached  to  the  rod  at  C;  for  if  ^  is 
the  total  angle  through  which  the  wheel  turns 
during  the  motion,  s  =  rd,  where  r  is  tlie  radius 
of  the  wheel,  and  0  is  measured  in  radians. 
Hence  A  =  ls  =  Ird  ;  the  value  of  6  is  read  off 
from  a  dial  attached  to  the  wheel ;  I  and  r  are 
known  lengths. 

In  Amsler's  polar  planimeter,  one  end  b  of  the  rod  ab  is  forced  to 
trace  once  around  a  given  closed  curve  whose  area  is  desired  ;  the  other 


Fig.  56 


246 


APPROXIMATION 


[VIII,  §  128 


end  a  is  mechanically  forced  to  move  back  and  forth  along  a  circular 
arc  by  being  hinged  at  a  to  another  rod  Oa,  which  in  its  turn  is  hinged 
to  a  heavy  metal  block  at  0.  As  h  describes  that  part  of  the  given  curve 
which  lies  farthest  from  0,  the  rod  ab  sweeps 
over  an  area  between  the  circular  arc  traced 
by  a  and  the  outer  part  of  the  given  curve  ;  as 
h  describes  the  part  of  the  curve  nearest  to  0, 
ab  sweeps  back  over  a  portion  of  the  area  cov- 
ered before,  between  the  circle  and  the  inner 
part  of  the  given  curve.  This  latter  area  does 
not  count  in  the  final  total,  since  it  has  been 
swept  over  twice  in  opposite  directions.  Hence 
the  quantity  A  —  Ird,  given  by  the  reading  of 
the  dial  on  the  machine,  is  precisely  the  desired 
area  of  the  given  closed  curve,  which  has  been 
swept  over  just  once  by  the  moving  rod  ab. 
In  practicing  with  such  a  machine,  begin 
with  curves  of  known  area.  The  machine  is  useful  not  only  in  finding 
areas  of  irregular  curves  whose  equations  are  not  known,  but  also  in  check- 
ing integrations  performed  by  the  standard  methods,  and  in  giving  at  least 
approximate  values  for  integrals  whose  evaluation  is  difficult  or  impossible. 
For  further  information  on  integrating  devices,  see  :  Abdank-Abaka- 
nowicz,  Les  integraphes  (Paris,  Gauthier-Villars)  ;  Henrici,  Rej)ort  on 
Planlmcters  (British  Assoc.  1894,  pp.  496-52.3);  Shaw,  Mechanical  Inte- 
grators (Proc.  Inst.  Civ.  Engs.  1885,  pp.  75-143);  Encyclopadie  der 
Math.  Wiss.,  Vol.  II.  Catalogues  of  dealers  in  instruments  also  contain 
umch  really  valuable  information. 


Fig.  57 


129.  Tabulated  Integral  Values.  —  Another  method  of  ob- 
taining the  values  of  certain  integrals  is  to  look  them  up  in 
numerical  tables  which  have  been  calculated  by  the  foregoing 
methods  or  by  other  means.  In  these  tables  are  printed  the 
values  of  the  integral  I: 


(1) 


•=r 


f(x)dx, 


where  a  is  some  convenient  constant,  for  a  large  number  of 
values  of  the  (variable)  upper  limit  u  which  differ  by  small 
amounts. 


VIII,  §  129]      APPROXIMATE   INTEGRATION  247 

Thus  tables  of  cominou  logarithms  are  precisely  tables  of 
values  of  the  integral 

(2)    I  =  J^J^^^dx=log,oe[\og,x~r'^=\og,,x'Y'^=^^Sio^- 


Among  other  integrals  which  may  be  found  thus  tabulated  are 
the  inverse  hyperbolic  sine  (see  Tables,  Y,  C)  : 


(3) 


/=  I        —    '        =  log  («  +  V«^  +  1)  =  sinh"^  u  5 
»/.^    Var'  + 1 


the  elliptic  integral  of  the  first  kind  (see  Tables,  Y,  D) 


-=«   V(l-a'^)(l-A;2ar')     »^a=o    Vl-Aj-sin^a 

where  x  =  sin  «  and  m  =  sin  <^ ;  and  others  which  are  defined 
in  the  Tables,  Y,  E-H.  Many  other  integrals  can  be  re- 
duced to  these,  just  as  many  integrals  can  be  expressed  in 
terms  of  logarithms. 

These  tables  give  corresponding  values  of  the  integral  I  and 
its  upper  limit  u ;  hence  they  define  7as  a  function  of  u : 

that  is,  the  integral  is  a  function  of  its  (variable)  iqyj^er  limit. 

The  values  of  these  integrals  can  be  read  off  also  from  a 
properly  constructed  graph  in  which  their  values  are  plotted 
in  the  usual  manner.  Thus  the  curve  u  =  sinh  /  (see  Tables, 
III,  E)  may  be  used  to  obtain,  approximately,  the  value  of  / 
when  7c  is  given ;  that  is,  the  values  of  the  integral  of  equation 
(3)  for  given  values  of  u. 


248  APPROXIMATION  [VIII,  §  129 

EXERCISES  LII.— ENTEGRATING  DEVICES     NUMERICAL  TABLES 

1.  Construct  a  figure  of  each  of  the  types  mentioned  below,  with  di- 
mensions selected  at  random,  and  find  their  areas  approximately  by  count- 
ing squares  ;  by  Simpson's  rule  ;  by  the  planimeter,  if  one  is  available. 
(a)  A  right  triangle,  (b)  An  equilateral  triangle,  (c)  A  circle,  (d)  An 
ellipse.  (Draw  it  with  a  thread  and  two  pins.)  (e)  An  arch  of  a  sine 
curve.     (/)  An  arch  of  a  cycloid. 

2.  Evaluate  the  integrals  below  approximately,  by  drawing  the  graphs 
of  the  integrands. 

(a)    r  Vl  +  x*  dx.  (d)    i    &^dx.  (g)    \     sine-^dx. 

(6)  ^^'^''sinx^dx.  (e)    £'e-^^dx.  (h)  |Vl^dx. 

((■)     y  s'mVxdx.  (/)    y  ^/smxdx.  (i)     \'^e^"^''dx. 

3.  The  integral  of  y^  from  .r  =  a  to  x  =  b,  divided  by  b  —  a,  is  called 
the  mean  square  ordinate  of  the  curve  y  =zf{x)  from  x  =  a  to  x  =  6. 
Find  the  mean  square  ordinate  for  the  curve  y  =  sin  x,  both  by  actual  in- 
tegration and  by  approximate  methods.     See  Ex.  33,  p.  226. 

4.  Show  that  the  mean  square  ordinate  oi  y  =  k  sin  x  may  be  found 
graphically  by  plotting  the  circle  p  =  k  sin  d  in  polar  coordinates,  since 

Ic^  \  ^  sin'-^  6  dd  is  twice  the  area  of  this  circle. 

5.  Show  in  general  that  the  mean  square  ordinate  of  y  —f{x)  can  be 
found  graphically  from  the  polar  figure  for  p  =f{d). 

6.  In  the  theory  of  electricity,  it  is  shown  that  the  effective  electromo- 
tive force  ^  of  a  current  is  the  square  root  of  the  mean  square  of  the 
actual  (variable)  electromotive  force  e.  Use  the  method  of  Ex.  5  to  find 
^  if  e  =  100  sin  6  +  10  sin  2  0,  from  ^  =  0  to  ^  =  tt,  where  6  is  the  angle 
described  by  certain  moving  parts  of  the  generating  machinery. 

7.  Proceed  as  in  Ex.  6  for  the  following  experimental  values  of  e; 
from  e  =  0  to  ^  =  90°  : 


0° 

]()° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

OO'' 

3 

50 

CA 

52 

58 

107 

130 

150 

165 

145 

VIII,  §  129]      APPROXIMATE  INTEGRATION 


249 


8.  The  figures  below  are  repro- 
ductions of  indicator  cards,  taken 
from  tliree  different  types  of  en- 
gines. Tlie  dotted  curves  are  en- 
tirely separate  from  the  full  lines. 
The  average  pressure  on  the  piston 
is  the  area  of  one  of  these  curves 
divided  by  the  length  of  stroke. 
Find  this  value  in  each  case,  where  the  stroke  is  12  in.  in  the  first  figure, 
and  8  in.  in  each  of  the  others.     (Unit  of  area  =  1  large  square.) 

[Note.    The  xoork  done  is  precisely  the  area  in  question,  on  a  proper 
scale,  since  the  work  is  the  average  pressure  times  the  length  of  stroke.] 


■r-i  -pi -pi  T 

TT" 

W*^!  '  '  1 

1     1 

lb"        1  :   1 

M  : 

i  -  ii42t^ 

M— 

-— .U-kr-" 

jiT- ,,.(!  . 

-TT      ^^ 

.i  ^■-"U- 

■" 

1 

-I-- 

-iM=^^^ 

1  L-i^jju-^ 



i        1 — r 

"h — tnT" 

i 

i  1  1  M    1 

T^l                                               "    ~             . 

i    V,                           :            : 

1             s 

1     0  "6 1  rr4-  J 

1  i^    ::::=^-=.-..--      

9.  A  piece  of  land  lies  between  a  straight  road  and  a  river  which 
crosses  the  road  at  two  points.  The  perpendicular  distances  from  road  to 
river  at  intervals  of  20  yd.  are  0,  15,  35,  40,  50,  45,  45,  30,  20,  10,  5,  0  yd. 
Find  approximately  the  area  of  the  land  by  each  of  the  methods  described 
above. 

10.  Find  from  the  tables  the  values  of  each  of  the  following  integrals  : 


(e)    J^      Vl  -  .O^sin^ede. 


<''  xif--     «  m: 


Vl  -  .09sin2^ 


(^)X 


1/2^1^.  49  .t2 


dx. 


<">  Z': 


>     V(l 


x-^)(l 


tiZ^) 


Vl  —  .64  sin2  9 
(0    J^'e-'x-erfx. 


250 


POLYNOMIAL  APPROXIMATIONS     [VIII,  §  129 


11.  Reduce  each  of  the  following  integrals  to  standard  forms  to  be 
found  in  the  tables  by  means  of  the  substitutions  indicated,  and  then 
evaluate  them : 


a)    C"'  Vl  -  .09  cos-^edd; 


dd 


V'l-.49sin2  2^ 
dx 


'^    X  logx' 
i)    \      e    idx; 


dx 


\/(4-a;2)(4-  .36x2) 


put  e  =  90°  —  xp. 
put  2  ^  =  i^. 
put  log  X  =  u. 


put  -  =  M. 


PART   II.     POLYNOMIAL   APPROXIMATIONS 
SERIES      TAYLOR'S   THEOREM 

130.   Rolle's  Theorem.     Let  us  consider  a  curve 

where  f(x)  is  single-valued  and   continuous,  and  where   the 

curve  has  at  every  point 
a  tangent  that  is  not  ver- 
tical. If  such  a  curve  cuts 
the  a;-axis  twice,  at  a;  =  a 
and  x  =  b,  it  surely  either 
has  a  maximum  or  a  mini- 
mum at  at  least  one  point 

x  =  c  between  a  and  b.     It  was  shown  in  §  39,  p.  64,  that  the 

derivative  at  c  is  zero : 

[A]    If    /(a)  =/(&)=  0,    tJien  \*^f(-^U    =    o,     (a<c<6); 


Fig.  5.S 


this  fact  is  k 


IS  Known 


as  Rolle's  Theorem. 


VIII,  §131]    INCREMENTS  — LAW  OF  THE  MEAN      251 

131.  The  Law  of  the  Mean.  Rolle's  Theorem  is  quite  evi- 
dent  geometrically  in  the  form :  A)i  arc  of  a  simple  smooth 
curve  cut  o^ff  by  the  x-axis  has  at  least  one  horizontal  tangent. 
The  precise  nature  of  the 
necessary  restrictions  is 


'y=fM 
f(t>)-f(a) 


Fig.  59 


given  in  §  130. 

Another  similar  state- 
ment, which  is  true  under 
the  same  restrictions  and 
is  equally  obvious  geo- 
metrically, is :  An  arc  of 
a  sinqjle  smooth  curve  cut  off  by  any  secant  has  at  least  one 
tangent  iKirallel  to  that  secant. 

If  the  curve  is  y  =f(x),  and  if  the  secant  yiS  cuts  it  at  points 
P:  [rt, /(tt)]  and  Q:  [b,  f{b)'],  the  slope  of  S  is 

A^-Ax=[/(6)-/(a)]-(&-a). 

The  slope  of  the  tangent  CT  at  a;  =  c  is  equal  to  this : 

{a<c<b). 


^-^  Ct  "'^1= 


f{b)-f{a) 
b  —  a 


A?/ 


This  statement  is  called  the  Law  of  the  Mean  or  the  Theo- 
rem of  Finite  Differences. 


It  is  easy  to  prove  this  statement  algebraically  from  Rolle's  Theorem. 
For  if  we  subtract  the  height  of  the  secaut  ,S'  from  the  height  of  the  curve, 
we  get  a  new  curve  whose  height  is : 


L      b  —  a 

Now  D  (x)  is  zero  when  x  =  a  and  when  x  - 
dZ>(x)/dx  =  Oatx  =  c,  {a<c<b): 


-a)  +  /(a)]- 

b.     It  follows  by  §  130  that 


dx    ]x=c     \    dx        L      b  -  a 
■which  is  nothing  but  a  restatement  of  [i?]. 


=  0, 


(«<c<6), 


252  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  132 

132.    Increments.     The  law  of  the  mean  is  used  to  determiue 
increments  approximately,  and  to  evaluate  small  errors. 
If  y  =f{x)  is  a  given  function,  we  have,  by  §  131, 

In  practice  this  law  is  used  to  estimate  the  extreme  limit  of 
errors,  that  is,  the  extreme  limit  of  the  numerical  value  of  A?/. 
It  is  evident  that 

[15*]  |A//l^Jfi.|Acc|, 

where  M^  is  the  maximum  of  the  numerical  value  of  dy/dx  be- 
tween a  and  a  +  Ax.  When  Ax  is  very  small,  the  slope  dy/dx 
is  practically  constant  from  a  to  «  +  Ax  in  most  instances,  and 
M-i^  is  practically  the  same  as  the  value  of  dy/dx  at  any  point 
between  a  and  a  -f  Ax. 

Example  1.  To  find  the  correct  increments  in  a  five-place  table  of 
logarithms. 

The  usual  logarithm  table  contains  values  oi  L  =  logio  N  at  intervals 
of  size  AiV  =  .001.     Hence 

where  N'<,c<N  +  .001. 

Logarithms  are  ordinarily  given  from  iV=  1  to  iV=  10.  Hence  Ai 
will  vary  from  .00043  at  the  beginning  of  the  table  to  .00004  at  the  end 
of  the  table.  This  agrees  with  the  "differences"  column  in  an  ordinary 
logarithm  table,  ' 

Example  2.  The  reading  of  a  certain  galvanometer  is  proportional  tq^^ 
the  tangent  of  the  angle  through  which  the  magnetic  needle  swings. 
Find  the  effect  of  an  error  in  reading  the  angle  on  the  computed  value  of 
the  electric  current  measured.     We  have 

C  =  k  tan  e, 

where  C  is  the  current  and  6  the  angle  reading.  Hence  the  error  Ec  in 
the  computed  current  is 

^c  =  AC  =  ^^'^^"^1  ^^  =  kM ■  &e(^^ a,  {e<a<e  ±Ad-), 


N=c 


VIII,  §132]     IXCREMEXTS  — LAW  OF  THE   MEAN       253 

where  Ec  is  the  error  in  the  computed  value  of  the  current,  and  M  is 
the  error  made  in  reading  the  angle  6.  Since  A^  is  very  small, 
Ec=k  sec^  e  •  A^,  approximately.  The  error  Ec  is  extremely  large  if  6  is  near 
90°,  even  if  Ad  is  small ;  hence  this  form  of  galvanometer  is  not  used  in 
accurate  work. 

EXERCISES   Lin.  — INCREMENTS      LAW    OF   THE   MEAN 

1.  At  what  point  on  the  parabola  y  =  x^  is  the  tangent  parallel  to  the 
secant  drawn  through  the  points  where  x  =  1  and  z  —  2? 

2.  Proceed  as  in  Ex.  1  for  the  curve  y  =  sin  x,  and  the  points  where 
a;  =30°  andx  =31°. 

3.  Proceed  as  in  Ex.  1  for  the  curve  y  =  log  (1  +  x),  for  x  =  .5  and 
«  =  .6. 

4.  Discuss  the  differences  in  a  four-place  table  of  natural  sines,  the 
argument  interval  being  10'. 

5.  Proceed  as  in  Ex.  4  for  a  similar  table  of  natural  cosines ;    of 
natural  tangents. 

6.  Discuss  the  differences  in  a  four-place  table  of  logarithmic  sines, 
the  entries  being  given  for  intervals  of  10'. 

7.  Proceed  as  in  Ex.  6  for  a  table  of  logarithmic  tangents. 

8.  Calculate  the  difference  in  a  seven-place  table  of  logio  sin  x  at  the 
place  where  x  =  30°  ;  where  x  =  60°  ;  where  x  =  85°. 

9.  Discuss  the  effect    of    a    small   change    in    x    on    the  function 
y  =  log  (1  +  1/x). 

10.  If  logioiV=  1.2070  ±  .0002,  what  is  the  uncertainty  in  N?  [The 
tei-m  ±  .0002  indicates  the  uncertainty  in  the  value  1.2070.] 

11.  If  the  angle  of  elevation  of  a  mountain  peak,  as  measured  from  a 
point  in  the  plain  5  mi.  distant  from  it,  is  5°  20'  ±  2',  what  is  the  un- 
certainty in  the  computed  height  of  the  peak  ? 

12.  The  horizontal  range  of  a  gun  is  i?  =  (I'V?)  sin  2a,  where  V\s 
the  muzzle  speed  and  a  the  angle  of  elevation  of  the  gun.  If  F=  1200 
ft. /sec,  discuss  the  effect  upon  R  of  an  error  of  5'  in  the  angle  of  elevation. 

13.  The  distance  to  the  sea  horizon  from  a  point  h  ft.  above  sea  level 
is  D  =  V'2  Rh  +  h-,  where  R  is  the  radius  of  the  earth.  Discuss  the 
change  in  D  due  to  a  change  of  one  foot  in  h.  (Z>,  R,  and  h  are  all  to 
be  taken  in  the  same  units.)  If  D  is  tabulated  for  values  of  h  at  inter- 
vals of  one  foot,  what  is  the  tabular  difference  at  the  place  where  A  =  60  ? 


254  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  132 

14.  If  the  boiling  point  of  water  at  height  H  ft.  above  sea  level  is  2', 
if  =  517  (212°  -  T)  -  (212°  -  T)^,  T  being  the  boiling  temperature  in 
degrees  F.  Discuss  the  uncertainty  in  H,  if  T  can  be  measured  to  1°. 
If  H  be  tabulated  with  argument  T  at  intervals  of  1°,  what  is  the  tabular 
entry  and  the  tabular  difference  when  T  =  200°  ? 

15.  When  a  pendulum  of  length  I  (feet)  swings  through  a  small  angle 
a  (radians),  the  time  (seconds)  of  one  swing  is  T  =  tt  VT/g  (1  +  a^/16). 
What  is  the  effect  on  T  of  a  change  in  a,  say  from  5°  to  6°  ?  Of  a  change 
in  I  from  36  in.  to  37  in.  ?     Of  a  change  in  g  from  32.16  to  32.2  ? 

16.  The  viscosity  of  water  at  6°  C.  is  P  =  1/(1  +  -0337  0  +  .00022  ff^). 
Discuss  the  change  in  P  due  to  a  small  change  in  d.  What  is  the  average 
value  of  P  from  ^  =  20°  to  ^  =  30°  ? 

17.  The  quantity  of  heat  (measured  in  calories)  required  to  raise  one 
kgm.  of  water  from  0°  C.  to  0°  C.is  H  =  94.21  (365  -  6i)0-3i25  +  j^.  How 
much  heat  is  required  to  raise  the  temperature  of  one  kgm.  of  water  1°  C. 
when  e  =  10°?    20°?    30°?    70°?    To  find  A;,  observe  that  lf=:0  when  ^=0. 

18.  The  coefficient  of  friction  of  water  flowing  through  a  pipe  of 
diameter  D  (inches)  with  a  speed  V  (ft. /sec.)  is  /=  .0126  +  (.0315 
—  .06  D)/\/V.     What  is  the  effect  on  / of  a  small  change  in  V?  in  D? 


19.    If  the  values  of    \    VI  — .2sin^xfto  were  tabulated  with  x  as 
Jo 
argument,  for  every  degree,  what  would  be  the  tabular  difference  at  the 
place  in  the  table  where  x  =  30°  ?     See  Tables,  V,  E. 

133.  Limit  of  Error.  In  using  the  formula  [B]  the  uncer- 
tainty in  the  value  of  c  is  troublesome.  If  the  value  of  dy/dx 
at  a;  =  a  is  used  in  place  of  its  value  at  a?  =  c,  the  error  made  in 
finding  Ay  by  [JB]  can  be  expressed  in  terms  of  the  second 
derivative  d'^y/dx'. 

We  shall  use  the  convenient  notation  mentioned  in  Ex.  33, 
p.  57,  and  Ex.  5,  p.  222,  for  the  derivatives  of  f(;x)  : 

f(x)  =  M^)    =  ^  (the  slope  of  2/  =  f(x)). 
cix  ctx 

y .(^)  =  ^m  =tE^^ <KM  (the  flexion). 


VIII,  §  133] 


ERROR  — TAYLOR'S  THEOREM 


255 


Let  M2  denote  the  inaxiinuin  of  the  numerical  value  oi  f"(x) 
between  two  points  x  =  a  and  x  =  h,  so  that 


(1) 


\n^)\^M,. 


The  area  under  the  curve  y=f\x)  between  x  =  a  and  any 
point  x=.x  between  a  and  h  is  evidently  not  greater  than  the 
area  under  the  horizontal  line 
y  =  M., ;  that  is,  if  a  <  a;  <  6, 


{2)\C^y%x)dxWC^\M, 

I V  x=a  I        *' x=a 

01-  |/'(aO?^^'|<J/,.rT' 


dx. 


since  df'ix)/dx=f"{x),  and  M.^ 
is  a  constant;  whence  substitut- 
ing the  limits  of  integration  in 
the  usual  manner : 

(3)    \f\x)-f\a)\^M,{x-a), 

which  is  geometrically  shown 

in  Fig.  60.     It  follows  that  the  Fig.  go 

area  under  the  curve  y  =f'{x)  —/'(«)  is  not  greater  than  that 

under  the  line  y  =  M2{x  —  a) : 


-Ll        -^_^^^    1          UK 

^t  „szte  ^fc 

7     ',        ^^     ?T~ 

t--^    V    "32^> 

.        \    ^J^  \l^ 

1^    ..       L^^-^ 

I  ^^     M    ti^. 

t  .zn    zB^n^  r 

_,  ...    \    w  ^^/ 

1        -4-.7a:\-^^l^ 

7    --S\w  V  y 

1  TTji/^    A_/ 

//T  /17 

"    .m  y  "± 

i/j^f/     / 

Aj¥^       / 

-'^cr^^    L 

-.    JZXIJ?'^        I 

- ;?    ?^  j?^          i 

0^0             X          0\   \ 

/L...       1 

(4) 


dx  \ 


or  since /'(a)  and  3/2  are  constants  and  df(x)/dx=f'{x), 


256  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  133 

whence,  substituting  the  limits  in  the  usual  manner, 

which  holds  for  all  values  of  x  between  x  =  a  and  x='b.  This 
formula  may  be  written  even  if  a;  <  a : 

^  [C*] /(ic)  = /(a)  + /'(a)(£c— a) +^2,  where  I  ^2 1  ^ -Ms^^^^^, 

and  En  is  the  error  made  in  using  f'(a)  in  place  of  /'(c)  in 
formula  [C]  ;  for  (x  —  a)^  =\x  —  a\'-. 

It  should  be  noticed  that  E2  is  exactly  the  error  made  in 
substituting  the  tangent  at  a;  =  a  for  the  curve,  i.e.  it  is  the 
difference  between  A?/ [=/(.«) —/(a)]  and  d?/[=/'(a)(a;— a)] 
mentioned  in  §  31,  p.  50,  and  shown  in  Fig.  12. 

The  formula  [-B*]  is  exactly  analogous  to  [C*] ;  since 
A//  =f(x)  — /(«)  if  Ax  =  a;  —  a,  [-B*]  may  be  written 

[5*]  f(x)=f{a)^E„  \E,\^M,.\x-a\. 

Example  1.     In  Ex.  1,  p.  252,  we  found  for  L  =  logio  N, 

AL  =  -^  (nearly). 

Applying  [C*],  with  f(N)  =  logio  iV,  a  =  N,  x  =  J^  +  AJST,  x—a  =  AN 
=  .001,  we  find 

AL=AN+AN)-fm=-^  +  E.,,  \E.J<-^.M,, 

where  I/2  is  the  maximum  value  of  |/"(2V)  |  =  (logio  e)/N'^  between  iV  = 
1  and  iV=  10.     Hence  E^  <  .00000022.     The  value  of   AL  found  before  x, 
was  therefore  quite  accurate,  — absolutely  accurate  as  far  as  a  five-place 
table  is  concerned. 

Example  2.  Apply  [C*]  to  the  function  f(x)  =  sin  x,  with  a  =  0,  and 
show  how  nearly  correct  the  values  are  for  x  <  ir/90  =  2°. 

Since  /(x)  =  sin  x,  and  a  =  0,  [  C*]  becomes 

sin  X  =  sin  (0)  +  cos  (0)  •  {x  -  0)  +  Eo  =  x  +  E2,     {Eol^M^j, 

where  J/o  is  the  maximum  of  |  /  "(x)  |  =  |  —  sin  x  |  between  0  and  tt/QO, 
that  is  M-z  =  sin  (tt/OO)  =  sin  2°  =  .0.349.     Hence  Eo  <  .0175x2.      Since 


VIII,  §  134]  ERROR  — TAYLOR'S  THEOREM  257 

a;<7r/90,  x2  <7rV8 100  <  .0013  ;   hence   ^2  <  .000023,  and  sin  a;  =  a;   is 
correct  up  to  x  ^  7r/90  within  .000023. 

Similarly,  for  a  =  7r/4,  we  have,  by  [C*], 


where  J/o  <  1-     If  (a;  -  7r/4)  <  tt/OO,  |  -F,  ;  <  (7r/90)2  h-  2  =  .0007. 


134.   Extended  Law  of  the  Mean.     Taylor's  Theorem.     The 

formula  [C*]  can  be  extended  very  readily.     Let  f\x),  f^\x), 
/'"(x),  •••  p"^x  denote  the  first  n  successive  derivatives  oi  fix) : 

-'     ^  '        dx"  dx      ' 

and  let  the  maximum  of  the  numerical  value  of  /^"^(x)  from 
jc  =  a  to  ic  =  6  be  denoted  by  J/„.     Then 

|/<")(.r)  I  ^  Jf„, 
and  I  f^f'Xx)  dx  |  ^  I    C^~' M„  dx\,  or 

|/("-i)(.r)  _/"-i)(a)  I  ^  I  M,,{x-a)\ 

for  all  values  of  x  between  a  and  h.     Integrating  again,  we 
obtain,  as  in  §  133  : 

|/(»-2)(^.)  _/(n-2)(a)  -fi^^\a)(x -  a)  I  <  1 3f/-^Hjzi^' | ; 

and,  continuing  this   process   by  integrations  until  Ave  reach 
f{x),  we  find : 

[I>]        \f(oc)-f(a)-f'(a)(oc-a)-^'^p-(x-ay-:. 


(M  — 1)1  I  Ml 


258  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  134 

or, 


where 


i«ni<M,.!i=fii:, 


and  where  J/"„  is  the  maximum  of  |  J^'^^(x)  \  between  x  =  a  and 
x  =  b. 

This  formula  is  known  as  the  Extended  Law  of  the  Mean,  or 
Taylor's  Theorem,  after  Taylor,  who  first  gave  such  approxi- 
mations as  it  expresses.     It  is  one  of  the  more  important  for- 
mulas of  the  Calculus. 
•  In  particular,  if  a  =  0,  the  formula  becomes 

[!>*]'  /(x)  =/(0)  +/'(0)x+^^:r2^  -. 
/(-i)(0)  , 

where  |  ^„  1  <  iJ/"„  |  .t"  |  /n !     This  special  case  of  Taylor's  Theo- 
rem is  often  called  Maclaurin's  Theorem. 

The  formula  [D*]  replaces  /(.«)  by  a  polynomial  of  the  nth 
degree,  with  an  error  E^.  These  polynomials  are  represented 
graphically  by  curves,  which  are  usually  close  to  the  curve 
which  represents  f(^x)  near  x  =  a.     See  Tables,  III,  K. 

Since  the  expression  for  £"„  above  contains  n !  in  the  denomi-^ 
nator,  and  since  n !  grows  astoundingly  large  as  n  grows  larger, 
there  is  every  prospect  that  E^  will  become  smaller  for  larger 
n ;  hence,  usually,  the  polynomial  curves  come  closer  and  closer 
to  f{x)  as  n  increases,  and  the  approximations  are  reasonably 
good  farther  and  farther  away  from  x  =  a.  But  it  is  never 
safe  to  trust  to  chance  in  this  matter,  and  it  is  usually  possible 
to  see  what  does  happen  to  E^  as  n  grows,  without  excessive 
work. 


VIII,  §134]  ERROR  — TAYLOR'S  THEOREM  259 

Example  1.  Find  an  approximating  polynomial  of  the  third  degree 
to  replace  sin  x  near  x  =  0,  and  determine  the  error  in  using  it  up  to 

X  =  7r/18  =  10^ 

Since  f(x)  =  sin  x  and  a  =  0,  we  have  f'(x)  =  cos  x,  f"{r)  =  —  sin  x, 
/"'(.t)  =  — cos  x,/i^ (.»•)  =  +  sinx,  whence /(O)  =  0, /'(O)  =  l,/"(0)  =  0, 
/'"(O)  =  -  1  ;  and  [Max.  \f"{x)\  ]  =  [Max.  |  sin  .r  |  ]  =  sin  10°  =  .1736, 
between  x  =  0  and  x  =  tt/IS  =  10°.     Hence 

sin  X  =  0  +  1  .  (X  -  0)  +  0  +  (-  1)  .  ^-—^  +  .^4  =  a^  -  1^  -!-  ^4, 

where  |  ^4  |<  (.1736)  •  .rV4  !  <  (.1736)   (tt/IS)*  -  4  !  <  .000007,    when  x 
lies  between  0  and  tt/IS. 

In  general,  the  approximation  gi'ows  better  as  n  grows  larger,  for 
|/<"'(.v)|  is  always  either  |  sin  .r  |  or  [  cos  x  |;  hence  M^  <  1,  and  \  E„\-^ 
x"/n  !  which  diminishes  very  rapidly  as  n  increases,  especially  if  x  <  1  = 
57°.3.     For  n  =  7,  the  formula  gives,  for  x  >  0, 

smx  =  x-$+'^  +  E„      \E^\<xy7\. 


EXERCISES  LIV.  — EXTENDED   LAW  OF  THE  MEAN 

1.  Apply  the  formula  (C*)  to  obtain  an  approximating  polynomial 
of  the  first  degree  for  tan  x,  with  a  =  0.  Show  that  the  error,  when 
I  X  I  <  ir/90,  is  less  than  .00003.  Draw  a  figure  to  show  the  comparison 
between  tan  x  and  the  approximating  linear  function. 

2.  Apply  [i)*]  to  obtain  an  approximating  quadratic  for  cosx,  with 
a  =  0.  Show  that  the  error,  when  |  x  |  <  tt/IO  is  less  than  (Tr/lO)^  -^  3  ! 
Draw  a  figure. 

3.  Apply  [D*y  to  obtain  an  approximating  cubic  for  cosx,  near 
X  =  0.  Hence  show  that  the  formula  found  in  Ex.  2  is  really  correct, 
when  I  X  I  <  tt/IO,  to  within  (ir/lO)*  -^  4!     Draw  a  figure. 

4.  Obtain  an  approximation  of  the  third  degree  for  sin  x  near  x  =  7r/3. 
Show  that  it  is  correct  to  within  (7r/10)*^4  !  for  angles  which  differ  from 
ir/3  by  less  than  tt/IO.     Draw  a  figure. 

5.  Obtain  an  approximation  of  the  first  degree,  one  of  the  second 
degree,  one  of  the  third  degree,  for  each  of  the  following  functions  near 
the  value  of  x  mentioned  ;  find  an  upper  limit  of  the  error  in  each  case 


260  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  134 

for  values  of  x  which  differ  from  the  vakie  of  a  by  the  amount  specified  ; 
draw  a  figure  showing  the  three  approximations  in  each  case  : 

(a)  e^   a  =  0,    !«- a  I  <  .1.  (e)  e-^  a  =  2,  |  a;  -  a  |  <  .5. 

(6)   tan  X,  a  =  0,  |  x  —  a  |  <  ir/90.         (/)  sin  x,  a  =  7r/2,  |  x— a  |  <  7r/45. 

(c)  log(l+x),  a=0,  |x-a|<.2.        {g)  tanx,  a=ir/4, 1  3^-«!1<t/90- 

(d)  cosx,  a=7r/4,  |x-a|  <7r/18.        (A)  x^+x  +  l,  a  =  l,  |x— a|  <  1/5. 
(0    2x2-x-l,  a  =  l/2,  |x-rt|<l. 

(;■)   x3  -  2  X-  -  X  +  1,  a  =  —  2,  I  X  -  a  I  <  .5. 

6.  Find  a  polynomial  which  represents  sin  x  to  seven  decimal  places 
(inclusive),  f or  |  x  |  <  10°. 

7.  Proceed  as  in  Ex.  6,  for  cos  x  ;  for  e-*,  when  0  <  x  <  1. 

8.  Show  that  x  differs  from  sin  x  by  less  than  .0001  for  values  of  x 
less  than  a  certain  amount ;  and  estimate  this  amount  as  well  as  possible. 

9.  Proceed  as  in  Ex.  8,  for  the  expressions  1  —  x"/2  and  cos  x. 

10.  Show  that  the  line  y  =  ax  +  b  which  passes  through  (0,  0)  and  has 
the  same  slope  as  y  =  sin  x  at  that  point,  is  precisely  the  same  as  the  re- 
sult of  formula  [C*]. 

11.  Show  that  1  —  x'^/2  agrees  with  cos  x  in  its  value,  its  first  deriva- 
tive, and  its  second  derivative,  at  x  =  0. 

12.  Express  log  (3/2)[=  log  (1  +  1/2)]  in  powers  of  (1/2)  so  that  the 
result  shall  be  correct  to  three  places. 

13.  What  is  the  maximum  error  in  the  approximation  x  sin  x  =  x^, 
when  \x[<  7r/12  ? 

14.  Show  that,  near  its  vertex,  the  catenary  y  =  cosh  x  has  nearly  the 
form  of  the  parabola  y  =  1  -h  x-/2.  Find  an  upper  limit  of  the  error  if 
|x|<0.1. 

15.  The  quantity  of  current  C  (in  watts)  consumed  per  candle  power 
by  a  certain  electric  lamp  in  terms  of  voltage  v  is  C  =  2.7+  I08007-.0767f. 
Express  C  by  a  polynomial  in  •!;—  115  correct  from  v  —  110  up  to  w  =  120 
to  within  .025  watt. 

135.   Application  of  Taylor's  Theorem  to  Extremes.    If  a 

function  y  —fix)  is  given  whose  maxima  and  minima  are  to  be 
found,  we  may  find  the  critical  points,  as  in  §  38,  p.  63. 
Let  a  be  one  solution  of  /'(.7j)  =  0,  that  is,  a  critical  value. 
Then,  since /'(a)  =  0,  we  have,  by  [-D*], 

Ay  =f{x)  -f{a)  =  0  +-^^  (0.  -  ay  +  E„\E,\<  M,  ^^j^', 


VIII,  §135]    TAYLOR'S  THEOREM  — EXTREMES  261 

where  J/g  ^  |/"'(^')1*  Heuce  tlie  sign  of  Ay  is  determined  by 
the  sign  of  /"(a)  when  (x  —  a)  is  sufficiently  small.  If /"(a) 
>  0,  Ay  >  0,  and  f(x)  is  a  minimum  at  a;  =  a  ;  if  /  "(a)  <  0,  Ay 
<  0,  and /(a;)  is  a  maximum  at  x  =  a.     (See  §  47,  p.  75.) 

If  f"(a)  =  0,  the  question  is  not  decided.*  But  in  that  case, 
by  [b*]  : 

Ay  =  f(x)  -/(a)  =  0  +  0  +  ^^^  (x  -  af  +-^  (x  -  «)"+  E„ 

where  |  ^5 1  <  3/5 1  x  —  a  1 75 !,  J/,^  |/^  (x)  \ .  From  this  we  see 
that  if/'"  (o)  9!=  0  there  is  neither  a  maximum  nor  a  minimum, 
for  (.X  — o)^  changes  sign  near  x  =  a.  But  if  f"'{a)  =0,  then 
/'''(a)  determines  the  sign  of  Ay,  as  in  the  case  of /"(o)  above. 
In  general,  if  /^*X")  i^  *^^^  fi^"^*-  °"®  °^  *^^®  successive  deriva- 
tives,/'(a), /"(a),  •••,  which  is  not  zero  at  x  =  a,  then  there  is  : 

no  extreme  if  k  is  odd ; 

a  maximum  if  k  is  even  and  /(^'^Ca)  <  0 ; 

a  minimum  if  A-  is  even  and  f^^^{fi)  >  0. 

Example  1.     Find  the  extremes  for  y  —  x*. 

Since  f(x)  =  :c*,  f'{x)=  4x3  ;  hence  the  critical  values  are  solutions  of 
the  equation  4  a:^  =  0,  and  therefore  x  =  0  is  the  only  such  critical  value. 

Since  /"(x)=12x2,  /"'(x)  =  24x,  /i^(a:)  =  24,  the  first  derivative 
which  does  not  vanish  at  x=:0  is /"'(a:),  and  it  is  positive  (  =  24).  It 
follows  that  /(x)  is  a  minimum  when  x  =  0;  this  is  borne  out  by  the 
familiar  graph  of  the  given  curve. 

EXERCISES  LV.  — EXTREMES 
1.    Study  the  extremes  in  the  following  functions  : 

(a)  x6.  (0    (.c  +  3)5.  (0  x2sinx. 

(6)  (x-2)8.  {f)  x\2x-iy.  (j)   x^cosx. 

(c)4x8-3x<.         {(j)    sinx8.  (A;)  x8  tan  x. 

(d)x8(l  +  x)8.         (;t)x-sinx.  (0    e-^^'^ 

*  Tlie  methods  which  follow  are  logically  sound  and  can  always  be  carried 
out  when  the  derivatives  can  be  found.  But  if  several  derivatives  vanish  (or, 
what  is  worse,  fail  to  exist) ,  the  method  of  §  40,  p.  64,  is  better  iu  practice. 


262  POLYNOMIAL  APPROXIMATIONS    Wm,  §  135 

2.  Discuss  the  extremes  of  the  curves  y  =  x",  for  all  positive  integral 
values  of  n. 

3.  Solve  the  problem  of  Ex.  18,  List  XIV,  p.  69,  by  the  method  of  §  135. 

4.  If  a  set  of  observed  values  of  a  quantity  y  which  depends  upon 
another  quantity  x  are  yo,  2/i,  2/2,  •■■■,  yni  when  x  has  the  values  .co,  cci, 
X2,  •■■,  Xni  and  if  y  is  connected  with  x  by  means  of  an  equation  of 
the  form  y  =  kx,  the  sum  of  the  squares  of  the  differences  between  the 
observed  and  the  computed  values  of  ?/  is  : 

S=iyo-  kxo)'  +  (2/1  -  ^•Xl)2  +  (1/-2  -  kxo^  +  •••  +  (?/„  -  ^x„)2. 

Show  that  the  sum  S,  as  a  function  of  k,  is  least  when 

2 xoiyo  -  kxa)  +'2xi  (yi  -  kxi)  +  •••  +  2 x„  (2/„  -  A;x„)  =  0, 

or  A;=2^^^.-2/.^2^,^-'^r. 

[Note.  Under  the  assumption  of  Ex.  18,  p.  69,  this  value  of  k  is  the 
best  compromise,  or  the  most  probable  value.] 

5.  Using  the  result  of  Ex.  4,  recompute  the  value  of  each  of  the  con- 
stants of  proportionality  k  in  Exs.  18-23,  p.  69. 

6.  An  open  tank  is  to  be  constructed  with  square  base  and  vertical 
sides  so  as  to  contain  10  cu.  ft.  of  water.  Find  the  dimensions  so  that 
the  least  possible  quantity  of  material  will  be  needed. 

7.  Show  that  the  greatest  rectangle  that  can  be  inscribed  in  a  given 
circle  is  a  square. 

[See  Ex.  25,  p.  70.  Other  examples  from  List  XIV  may  be  resolved 
by  the  process  of  §  135.] 

8.  What  is  the  maximum  contents  of  a  cone  that  can  be  folded  from 
a  filter  paper  of  8  in.  diameter  ? 

9.  A  gutter  whose  cross  section  is  an  arc  of  a  circle  is  to  be  made  by 
bending  into  shape  a  strip  of  copper.  If  the  width  of  the  strip  is  «,  show 
that  the  radius  of  the  cross  section  when  the  carrying  capacity  is  a  maxi- 
mum is  u/tt.  [Osgood.] 

10.  A  battery  of  internal  resistance  r  and  E.  M.  F.  e  sends  a  current 
through  an  external  resistance  E.  The  power  given  to  the  external 
circuit  is  „  2 

W=  —^ 

(B  +  ry 

If  e  =  3.3  and  r  =  1.5,  with  what  value  of  R  will  the  greatest  power 
be  given  to  the  external  circuit '?  [Saxelbv.] 


VIII,  §  136]  INDETERMINATE  FORMS  263 

11.  Find  the  shortest  distance  from  the  origin  to  the  curve  y  =  a'; 
show  that  it  is  measured  along  a  straight  line  from  the  origin  to  the 
intersection  of  the  given  curve  with  the  curve  x  =  —  y^  log  a. 

12.  Show  that  the  maximum  and  the  minimum  distances  from  a  point 
(rt,  b)  to  the  curve  y  =  x-  join  (a,  h)  to  the  intersections  oi  y  =  :r-  with 
x(y-  b  +  i)  =a. 

136.  Indeterminate  Forms.  The  quotient  of  two  functions 
is  not  defined  at  a  point  where  the  divisor  is  zero.  Such 
quotients  f(x)-r-<f}(x)  at  x  =  a,  where  f(a)  =  <l>(a)=0,  are 
called  indeterminate  forms.*     We  may  note  that  the  graph  of 

(1)  ^  =  :r7^'  (/(a.)  =  <^(a)  =  0), 

may  be  quite  regular  near  x  =  a;  hence  it  is  natural  to  make 
the  definition : 


(2) 


If  we  apply  [D  *],  we  obtain, 


,      f(x)      0+f'(a)(x-a)+E,' 
•d 

where 


^      <^(x)      0  +  <f>\a){x-a)+E2"' 


\E,'\^  M.:  (x  -  ay/2 1,  1  E,"  \  ^  MJ'  (x  -  ay/2\, 
and  J/2'  ^  I  /"  (x)  I,    M2"  >  I  <t>"(x)  I,  near  x  =  a. 

Hence  q  =  VH  = ^ 


*U  4>(a)=0  but  f(a)  ^  0  the  quotient  q  evidently  becomes  infinite ;  in  that 
case  the  graph  of  (1)  shows  a  vertical  asymptote. 


264  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  136 

where  p'  and  p"  are  numbers  between  —  1  and  + 1.     It  follows 

that 

(3)  hm  g  =  lim  ^;-(  =  ^yf^ 

unless  <j>'(a)  =  0.  But  if  ^'  (a)  =  0,  g  becomes  infinite,  and  the 
graph  of  (1)  has  a  vertical  asymptote  Sit  x  =  a  unless  /'  (a)  =  0 
also.  If  both  f  (a)  and  4>' (a)  are  zero,  it  follows  in  precisely 
the  same  manner  as  above,  that 

_f{x)       ^     ^  '^^       *^'  (A:  +  l)! 


^^^)      <^-(a)  +  y'3/Ui;^^ 


where  either /(*Xa)  or  >^^^\a)  is  not  zero,  but  all  preceding  de- 
rivatives of  both  fix)  and  ^{x)  are  zero  at  x  =  a\  and  where 
■^I+i  ^  1/'"+'^^;)  I,  iWlVi  ^  |'</.<^-+^>(ic)  I  near  x  =  a  and  where 
_p'  and  ^"  are  numbers  between  —  1  and  +  1.     It  follows  that 

lim  g  =  lim  •-——  =  -TTtVTT' 

provided  all  previous  derivatives  of  both  f(x)  and  <^  (x)  are  zero 
at  X  =  a,  and  provided  <^(*>  (a)  =^  0.  If  <^<*'  (a)  =  0,  /<*^  (a)  ^  0, 
then  g  becomes  infinite  and  the  graph  of  (1)  has  a  vertical 
asymptote  at  x  =  a. 

It  should  be  noted  that  (3)  is  only  a  repetition  of  Rule  [VII],  p.  36. 
For  if  u  -f{x)  and^;  =  (p{x),  since  /(a)  =  0(a)  =0, 

_  /(a:)  ^  /(■r)-/(ffl)  ^  4 m  ^  A«  _^  Au 
^~<f}{x)      4>{x)-<t>ia)      ^v      Ax  ■  Ax' 


where  Ax  =  x  —  a  ;  and  therefore 

,.      A?(      ,.      Atj     rd?<     dv-^      _r/!Ml      --^ 
S  '^  =  i^o  :^  "  i"o  ^x-\Tx-  TA^  -  L  0'(x)  J^.  -  0' 

provided  <?i'(a)  is  not  zero  (see  Theorem  D,  p.  18). 


r(a) 
(«)' 


VIII,  §  137]  INDETERMINATE  FORMS  265 

Example  1.     To  find  lira  [(taiix)-^  a;]. 

x  =  0 

Here  /(a:)  =  tan x,  <f>(x)  =  x;   /(O)  =  0(0)  =  0  ;  hence 
y^^  tan  X  _  f'(0)  ^  [sec2  xl^^  ^  ^ 

x^       X  0'(O)  1 

Draw  the  graph  5=(tana;)  — x  and  notice  that  this  value  (7  =  1  fits  ex- 
actly where  x  =  0. 

This  limit  can  be  found  directly  as  follows  : 

lim  ^^"  ^  ^  lim  ^^^  ^-^  +  ^^ ~  ^''"  '^^^  =  ^ ^^"  ^-1       =sec2x1      =1 
h^     h         b^        (0  +  /i)-(0)  dx    Jx=o  Jx=o 

Compare  the  work  done  in  §  96,  p.  167,  for  lim  (cos  A6  —  1)/A^. 
Example  2.     To  find  lim  (i  _  cos  x)/x2. 

Here  /(x)  =  1  -  cos  x,  0(x)  =  x2  ;  /(O;  =  <^(0)  =  0  ;  /'(O)  =  sin  (0)  =0 
and  ^'(0)  =  0  ;  /"(x)  =  cos  x,  0"(x)  =  2  ;  hence 


lim  h 


2    Jx=o~2" 


Draw  the  graph  of  g=(l  —  cosx)/x2,  and  note  that  (x  =  0,  q  =  1/2) 
fits  it  well. 

137.   Infinitesimals  of  Higher  Order.     When  the  quotient 

(1)  0=^ 

approaches  a  finite  number  not  zero  when  x  is  infinitesimal : 

(2)  \imq  =  \im^^  =  ]c=^0, 

^    ^  z  =  0  x  =  0       X" 

then  f(x)  is  said  to  be  an  infinitesimal  of  order  n  with  respect 
to  X.  An  infinitesimal  Avhose  order  is  greater  than  1  is  called 
an  infinitesimal  of  higher  order. 

The  equation  (2)  may  be  reduced  to  the  form 

(3)  lim[/(aj)-^u-"]=0, 

or 

(4)  fix)  =  (k^E)x'', 


266  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  137 

where  lim  E  =0.  The  quantity  A;x"  is  called  the  principal 
part  of  the  infinitesimal  /(x).  The  difference / (x)  —  kx"  =  Ex"" 
is  evidently  an  infinitesimal  whose  order  is  greater  than  «,  for 

lim  (Ex''  H-  x'^)  =  lim  ^  =  0. 

Thus  by  Example  2,  p.  265,  1  —  cos  x  is  an  infinitesimal  of  the  2d  order 
with  respect  to  x  ;  its  principal  part  is  x^/2.     Note  that 

1  —  cos  X  =  x'^/2  +  px^/3  !, 

by  [D*],  where  —  1  ^  p  ^  +  1 ;  the  principal  part  is  the  first  term  of 
Taylor's  Theorem  that  does  not  vanish. 

In  general,  if /(0)  =  /'(0)  =  /''(0)=  -  =/(^-i)(0)=0,  hutfW(0)^0, 
the  formula  [D  *]  gives,  for  a  =  0, 

/(a:)  =/W(0)  .  xk/k  \+pM„+iX^-+y(k  +  1)  ! 

where  M^+i  ^|/<*^+i*(a:)|  near  x  =  0,  and  —  l<p^  +  l.  Hence /(x)  is 
an  infinitesimal  of  order  k  with  respect  to  x,  and  its  principal  part  is 

fW{0)x''/kl 

EXERCISES  LVL— INDETERMINATE  FORMS      INFINITESIMALS 

1.  Evaluate  the  indeterminate  forms  below,  in  which  the  notation 
^(x)  I  a  means  to  determine  the  limit  of  0(x)  when  x  =  a.  The  vertical 
bar  applies  to  all  that  precedes  it.     Draw  the  graphs  as  in  Exs.  1,  2,  above. 

(a)  sin  x/x  |  q.  (b)  (tan  2  ■x)/x  \  g.  (c)  sin  ax/sin  bx  \  q. 


X  —  l|i  X     \o  X      \q  tan  3  x  |  o 

^  "      '  log(l+x)|o  Vx     lo 

-  /-«      •"  I  X  -  tan  X 1 0 

^^  ^  I  V    y   tan-ixlo 

log(x'-i-3)  I 
x^  +  3  X  -  10 1 2 ' 
sin-i(2  -  x)  I 
Vx^  -  3  X  +  2 1 2 
sin-i(Vffl''^-xVa)| 


(r) 


a'-b^\ 

X       \ 
logx 

0 

Vx2-1 
X  COS  X- 

1 
sinx 

X 

X 

logx 

sinx 
sin2x 

2  sin  X  - 

1 

jr 

-11 

X 

log 

X  o' 

e^- 

-  e^>°» 

X  — 

sinx 

C9)       °  •  (0 


(u) 


(s)    2smx-^  I  (^) 


sin  6  X     I  „/6  Va- 


VIII,  §  138]  INDETERMINATE   FORMS  267 

2.   Determine  the  order  of  each  of  the  quantities  below  when  the  vari- 
able X  is  the  standard  infinitesimal : 

(rt)  X  -  sin  X.  (e)   e  -  e^'n-^.  (i)   sin  2  x  —  2  sin  x. 

(6)  t'—e-'.  (/)  a'—\.  (j)  log  cos X. 

(c)  x2  sinx2.  {g)  log[(a+x)/(«-x)].  (^•)    log(l  +  e-Vx). 

^d)  log  (1  +  x)  —  X.      (ft)  X  cos  X  —  sin  x.  (?)     tan-i  x  —  sin-i  x. 


(w)  log  cos X— sin^ X.  («)  2x— e*+e~*.       (o)  cos-i(l— x)  — \/2x  — x^. 

3.  Show  that  Ex.  1  (a)  can  be  expressed  as  the  derivative  of  sin  x  at 
X  =  0,  as  in  Example  1,  p.  265. . 

4.  Show  that  Exs.  1  (e),  (/),  (/i),  (j)  can  be  expressed  as  the  deriva- 
tives of  the  numerators,  for  x  =  0. 

5.  Show  that  Ex.  1  (rZ)  can  be  expressed  as  the  derivative  of  its  numer- 
ator divided  by  the  derivative  of  its  denominator,  at  x  =  1. 

6.  Find  the  limit  of  the  ratio  of  the  surface  of  a  sphere  to  its  volume, 
as  the  radius  approaches  zero. 

7.  Find  the  limit  of  the  ratio  of  a  chord  of  a  circle  to  the  distance 
along  a  radius  perpendicular  to  the  chord  from  the  chord  to  the  circle. 

8.  Given  two  quantities  u  and  v  which  vary  with  the  time  «,  so  that 
u  =f(t)  and  V  =  0(0,  show  that 

lim  ^'=rilmf^M-|  lim"^ 


flim  ^'1 


9.  Show  that  the  slope  of  the  path  of  a  moving  body  is  the  ratio  of  its 
vertical  speed  to  its  horizontal  speed. 

138.  Double  Law  of  Mean.  Let  y  =/(x)  and  y  z=<p{x)  be  two 
simple  smooth  curves  between  x  =  a  and  x  =  6  ;  and  let  us  draw  the  two 
secants  which  cut  these  two  curves  at  the  points  x  =  a  and  x  =  b.  Then 
there  exists  a  point  c  such  that  the  ratio  of  the  slopes  of  the  curves  at  x  =  c 
is  equal  to  the  ratio  of  the  slopes  of  the  secants. 

^  <t>{b)-4>{a)       <!>'{€)' 

To  prove  this,  consider  the  parallel  curves 

y  =fi^)  -f(a)  and  J/  =  0  (x)  -  <f>  (a), 
which  both  go  through  the  same  left-hand  point  (a,  0)  and  the  ratio  of  the 
slopes  of  whose  secants  is,  as  above,  [/(6)  — /(a)]/[0  (&)  —  <P  (a)]- 


POLYNOMIAL  APPROXIMATIONS    [VIII,  §  138 


Multiplying  all  the  ordinates  of  y  =  cp^x)  —  4>{a)  by  this  ratio,  we 
have  the  new  curve 


<p(j3)-<p  (a) 


[0(x)-0(a)]  =  J'(a;),  (say). 

This  curve  has  both  the  same 
left-hand   point    (a,   0)    and    the 
right-hand  point 


m-fia) 


[6,  /(5)-/(a)] 
as  the  curve 


(p(b)-<p(a)  Hence,  by  RoUe's  Theorem,  there 
is  a  point  x  =  c,  for  which  the  two 
have  the  same  slope  ;  that  is,  such 


since  f(x)  —  F(x)  vanishes  a-t  x  =  a  and  x  =  b. 
above,  the  statement  (1)  is  proved. 


Replacing  F  by  its  value 


139.   The  Indeterminate  Form  oo  -^  oo.    Vertical  Asymptotes. 

Suppose  the  curves  y=f(x)  and  y  =  0  (x)  are  continuous  and  have  a 
continuous  slope  from  x  =  a  up  to 
but  not  including  x  =  A,  where 
both  ordinates  become  infinite ; 
that  is,  each  curve  has  a  vertical 
asymptote  at  x  =  A.  Then  b  and  a 
can  be  so  chosen  in  the  neighbor- 
hood of  A  that  both  f{a)/f{b)  and 
(t>{a)/4>  (b)  shall  be  as  small  as  we 
please.  For  however  close  to  A 
one  takes  a,  /(a)  and  0(a)  are 
finite.  Taking  now  b  between  a 
and  x4,  we  can  give  f{b)  any  value 
above  /(a).  Therefore  the  first  of 
the  preceding  ratios  (and  in  like 

manner  the  second)  can  by  proper  choice  of  6,  after  any  choice  of  a, 
made  as  small  as  one  pleases.  Notice  that  a  and  b  must  be  on  the  sa 
side  of  the  vertical  asymptote. 


Fig.  62 


be 


VIII,  §  140]  INDETERMINATE  FORMS  269 

Let  the  choice  of  a  and  b  be  made  as  just  indicated.     The  theorem  of 
§  138  still  holds  : 


0'(c)      0(6)-,^  (a)      fi      '^(^)lrf,(M 


(a<c<6), 


which,  by  the  preceding  remark,  approaches /(&)/0  (6)  as  6  approaches^. 
If  we  let  a  approach  A,  the  preceding  conditions  insure  that  c  and  b 
will  both  approach  A.    Thus,  finally, 


=,=^4  0'  (X)         r^A  (p  (X) 


that  is,  if /(«)  and  ^(x)  each  becomes  infinite  at  x  =  ^,  and  if  the  quo- 
tient/'(x)/(^'(x)  approaches  a  limit  as  x  =  A  from  either  side,  then 
/(x)/0(x)  approaches  the  same  limit. 

Similarly,  if/'  (x)/<p'  (x)  assumes  the  form  x  -f-  oo,  and  if /"(x)/0"(x) 
approaches  a  limit,  then  f'ix)/<p'(:r),  and  hence  also  /(x)/0(x) 
approaches  the  same  limit,  and  so  on. 

Since  /(x)/0(x)  =  [l/0(x)]  -i-  [l//(x)],  any  fraction  that  takes  one 
of  the  two  forms  0/0,  oo  -f-  oo,  can  also  be  put  into  the  other  form.  In 
practice  this  method  may  be  more  convenient  or  less  so,  than  the  preced- 
ing one,  depending  upon  the  particular  example.  Thus,  as  x  =  7r/2, 
tan  X  and  sec  x  both  become  infinite,  while  ctn  x  and  cos  x  approach  zero  ; 

hence 

tan  X       ,.       cosx      , 

lim   =   lim   =  i- 

I  £,^/  gSecx      li^/oCtnx 

140.   Other  Indeterminate  Forms.     Likewise,  if /(x)  =  0  as  .^(x) 

becomes  infinite,  their  product  is  of  the  form  0  x  oo,  and  it  can  be  put 
into  either  of  the  preceding  forms. 

Thus,  as  X  =  0,  log  x  becomes  —  oo  ;  so  that 

lim  (X  log  X)  =  lim  -^  =  lim  -^,  =  lim  (  -  x)  =  0. 

x  =  0  x  =  0     1/X  »  =  0-  1/x^         x  =  0 

Other  indeterminate  forms  are  oo  —  oo,  1°°,  0",  oo".     All  these  can  be 

made  to  depend  on  the  forms  already  considered.     For  let  a,  /3,  7,  5,  e, 

be  variables  simultaneously  approaching,  respectively,  00,  00,   1,  0,  0. 

Then  a  —  /3,  7a,  5«,  a«  take,  respectively,  the  preceding  four  indeterminate 

forms.     But  T/S_l/« 

lim  («-,)=  lim  IZ^, 


270  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  140 

•which  is  of  the  form  0/0  ;  while  the  logarithms  of  the  others, 

log7a  =  «log7,      log6<  =  elog5,     log  ««  =  e  log  «, 
are  each  of  the  form  0  x  oo. 

Example  1.     Thus,  when  x  =  7r/2,  (sin  x)t«°»  takes  the  form  1",     But 
(sin  a;)'^"^  =  e[iog6inx]/ctn.T^ 

which  approaches  the  same  limit  as  e-<=t°^/'=^=^*,  as  a;  =  7r/2,  and  this  limit 
is  evidently  e°  =  1. 

Example  2.     Similarly,  when  x  becomes  infinite,  (l/x)^/(2x+i)  takes 
the  form  O**.     It  may  be  written  in  the  form, 
e[-iogxi/(2i+i)^ 
which  approaches  the  same  limit  as  e'^/^i^  that  is,  the  limit  is  c"  =  1,  as 

X  =  00. 

Example  3.  As  an  example  of  the  last  form,  oo"^  take  (1/x)'  as  x  =  0. 
This  becomes  g-iiogx 

and  approaches  e"  =  1,  as  x  =  0. 

Indeterminate  forms  in  two  variables  cannot  be  evaluated,  unless  one 
knows  a  law  connecting  the  variables  as  they  approach  their  limits,  which 
practically  reduces  the  problem  to  a  problem  in  one  letter. 

EXERCISES  LVII.  —  SECONDARY  INDETERMINATE  FORMS 

1.  Evaluate  each  of  the  following  indeterminate  forms,  where  <^(x)|a 
means  the  limit  of  <p{x)  as  x  approaches  a.    Draw  a  graph  in  each  case. 

(«)  5|  •         ^^^  5| '         ^""^ ''"°''°' 

logctnxl      .         (ft)   12E^|   .  (n)    (l+x)V-|o. 

log  cos  X  l,r/2  ^      loo 

(c) ,  f""    I  .(0  1^1 .  (o)  (1  +  1/xrL. 

l0g(7r/2-X)   l,r/2  Z       loo 

(d)   -I   .  (j)  xctnxlo.  {p)    (tanx)=°"|,r/2. 

e^  L 

,  .    log  cos  X  I  ,  (jfc)   3-2  log  xs  I, .  (?)     (Sin  x)""'  lo . 

sin^x    lo 

(/)  l5?_?!!li^|  .  Q)  (tanx-secx)  L ,,.  (r)  ftanx ^ — "^1     • 

^•^^  logtanxlo  ^'  ^  '''^-   ^      V  T/2-xjl/^ 


VIII,  §  141]  INFINITE   SERIES  271 

2.  If  (p(:c)  =  2  —  2  cosh  x  +  x  sinh  .r,  show  that 

[0'(a:)/0(a-)]  L  =  oo  and  [0'(.c)/0(x)]  ^  =  L 

3.  Find  the  limit,  as  x  becomes  infinite,  of  the  product  x''e-'"  for  any 
positive  integral  value  of  n.  Draw  the  graphs  for  the  cases  n  =  1 ,  2,  3. 
Hence  show  that  the  damping  factor  g-^  reduces  the  curve  y  =  x"  to  a 
new  curve  asymptotic  to  the  x-axis. 

4  Show  that  the  improper  integral  of  xe~^  from  0  to  oo  exists,  and  that 
its  value  is  1.     [See  Tables,  IV,  97  a,  109  ;  and  V,  F.] 

5.  Show  that  x"*,  where  m  is  fractional,  lies  between  two  integral 
powers  of  x.  Hence  show  that  the  curve  y  =  x^e-^  is  asymptotic  to  the 
X-axis, 

6.  Show  that  the  improper  integral  of  x"'e-*  from  x  =  0  to  x  =  oo, 
where  m  is  any  positive  fraction,  exists,  by  use  of  Ex.  5. 

7.  Find  the  value  of  each  of  the  following  improper  integrals  from 
Table  V,  F: 

(1)  Cx'^e-^dx.  (3)    Cxe-'-^dx.  (5)    C  x^'^^e-^  dx. 

(2)  r*xO-2e-Mx.  (4)    Cx^-^e^dx.  (6)    C  x^-^e-^dx. 


8.   Show  that  the  derivative  of  log  x  is 
d  log  X  _  ,.     log  (x  +  ft)  —  log  X 


lim 


dX  A=M)  h 


■isG'-K'^')] 


=  -lim  log  (1  +  z)  ^'',  where  z  =  — 

X  z^  X 

Hence  show,  by  use  of  Ex.  1  (71),  that  the  derivative  m  question  is  1/x. 
9.    Throw  the  expression  of  Ex.  8  into  the  form 

llimr^log(l  +  ^)]  =  himi^, 
X  A=M)  La        \        x/ J     X  uii  w  —  1 

where  m  =  1  4-  h/x.    Show  that  the  last  limit  above  is  equal  to  1  ;  hence 
verify  the  result  of  Ex.  8. 

141.   Infinite  Series.     An  infinite  series  is  an  indicated  sum 
of  an  unending  sequence  of  terms  : 

(1)  ao-f-ai  +  ao+---+a„+ •••; 


272  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  141 

this  has  no  meaning  whatever  until  we  make  a  definition,  for 
it  is  impossible  to  add  all  these  terms.  Let  us  take  the  sum 
of  the  first  n  terms  : 

Sn  =  «o  +  «!  +  as  H H  a„_i, 

which  is  perfectly  finite  ;  if  the  limit  of  s„  exists  as  w  becomes 
infinite,  that  limit  is  called  the  sum  of  the  series  (1) : 

(2)  <S=  lira  s„  =  «o  +  ai  +  -- +««  +  ••• 

n=cK> 

If  lim  s^  =  S  exists,  the  series  is  called  convergent ;  if  S  does 

not  exist,  the  series  is  called  divergent;  if  the  series  formed 
by  ticking  the  numerical  (or  absolute)  values  of  the  terms  of 
(1)  converges^  then  (1)  is  called  absolutely  convergent.  Infi- 
nite series  which  converge  absolutely  are  most  convenient  in 
actual  practice,  for  extreme  precaution  is  necessary  in  dealing 
with  other  series.     (See  §  143,  p.  276.) 

Example  1.  The  series  1  +  r  +  r-  +  •••  +  r"  +  •••  is  called  a  geometric 
series ;  the  number  r  is  called  the  .common  ratio.  A  geometric  series 
converges  absolutely  for  any  value  of  r  numerically  less  than  1 ;  for 

s„  =  1  +  r  +  r2  +  -  +  r»-i  =  -J—  -  -^^  , 
1  -^  ;•      1  —  r 

hence  liml— ^i s„|  =  lim  |-^|  =  0,  if  |  r|  <  1,        ; 

since  r"  decreases  below  any  number  we  might  name  as  7i  becomes  in- 
finite.    It  follows  that  the  sum  ^S"  of  the  infinite  series  is 

8  =  lim s„  =  -i— ,  if  I r |<  1  ; 

and  it  is  easy  to  see  that  the  series  still  converges  if  r  is  negative,  when  r 
is  replaced  by  its  numerical  value  \r\. 

Example  2.  Any  series  ao  +  «i  +  as  +•••  +  ««  +  •  -of  positive  num- 
bers can  be  compared  with  the  geometric  series  of  Ex.  1.     Let 

(r„  =  ao  +  «!  +  ao  +  •••  +  a„-i ; 


VIII,  §  142]  TAYLOR  SERIES  273 

then  it  is  evident  that  (7-„  increases  with  n.  Comparing  with  the  geometric 
series  a^{\  +  r  +  r-  +  •••  +  r»  +  •••),  it  is  clear  that  if 

■where  s„  =  1  +  r  +  •••  +  »""~^.  Hence  o-„  approaches  a  limit  if  s„  does, 
i.e.  ?/  0  <  ?•  <  1.  It  follows  that  the  given  series  converges  if  a  value  of 
r  <  1  can  be  found  for  which  «„  ^  agr",  that  is,  for  which  a„  -r-  a„_i  <  r 
<1.  There  are,  however,  some  convergent  series  for  which  this  test  can- 
not be  applied  satisfactorily.  It  may  be  applied  in  testing  any  series  for 
absolute  convergence ;  or  in  testing  any  series  of  positive  terms.  For 
example,  consider  the  series 

1!2!3!  71  !  ' 

here  a„  =  1/n  !,  a„_i  =  l/(n  —  l)!i  and  therefore  a„/an-i  =  (n  —  1)  \/n  I 
=  1/n.     Hence  «„/"«-!  <  V^  when  re  ^  2, 

=  1  +  s„_i, 

where  s„_i  =  1  +  r  +  •••  +  r''-^,  r  =  1/2.  It  follows  that  the  given  series 
converges  and  that  it^  sum  is  less  than  1+2  =  3.  [Compare  §  143, 
p.  278  ;  it  results  that  e  <  3.     Compare  Ex.  2,  p.  275.] 

142.  Taylor  Series.  General  Convergence  Test.  Series  which 
resemble  the  geometric  series  except  for  the  insertion  of  con- 
stant coefficients  of  the  powers  of  r, 

(1)  A  +  Br-hCr  +  D)^-\--, 

arise  through  application  of  Taylor's  Theorem  [Z)*](§  134,  p.  258); 
such  series  are  called  Taylor  series  or  power  series.  The  prop- 
erties of  a  Taylor  series  are,  like  those  of  a  geometric  series, 
comparatively  simple.  Comparing  (1)  with  [i5*],  we  see  that 
r  takes  the  place  of  (x  —  a),  while  A,  B,  C,  D,  "•  have  the 
values : 

A=m,  B^fM,  c^-CM,  D^qf-. 


274  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  142 

If  we  consider  the  sum  of  n  such  terms : 

we  see  by  [Z)*],  that 

f(x)=s^  +  E^,  where  |^„|<  lf„  1^-=-^,  M„>\p-\x)\', 
or  s„  =f{x)  -  E^. 

It  follows  that  if  En  approaches  zero  as  n  becomes  infinite, 
the  infinite  Taylor  Series 

converges,  and  its  sum  is  ,S  =  lim  s„  =  fix).* 

This  is  certainly  true,  for  example,  whenever  |/^"'(cc)  |  remains, 
for  all  values  of  n,  less  than  some  constant  C,  however  large, 
for  all  values  of  x  between  x=  a  and  x=  b.     For  in  that  case 

lim|^J<lim(7»l'^'~^l"=(71iml^'~'^l"  =  0, 
„=«,  =  «=«  n !  n=fcoo      n ! 

for  all  values  of  (x  —  a).1[  When  I/^"'(^)  I  grows  larger  and 
larger  without  a  bound  as  n  becomes  infinite,  we  may  still 
often  make  |  E„  |  approach  zero  by  making  (x  —  a)  numeri- 
cally small. 

Example  1.  Derive  an  infinite  Taylor  series  in  powers  of  x  for  the 
function  f(x)  =  sin  x. 

Since /(x)=:  sin  x,  we  have  f{x)=  cosx,  f'(x)  =  —  sin  a;,  and,  in  gen- 
eral,   /("'(x)  =  ±  sinx,  or  ±cosx;  hence 

|/»(x)|<l,   liml^„|<lim  — =  0; 

—         >i=oo  ~-  n^=«  n ! 

*  This  result  is  forecasted  in  §  134,  p.  258. 

t  This  results  from  the  fact  that  n  eventually  exceeds  (a;  — a)  numerically; 
afterwards  an  increase  in  n  diminishes  the  value  of  En  more  and  more  rapidly 
as  n  grows. 


VIII,  §  142]  TAYLOR  SERIES  275 

therefore  the  infinite  series  [Z)**]  for  a  =  0  is 

sina;  =  0  + x  +  O -~a;3  +  0  + —  x^  +  ...  ; 
3 !  5  ! 

this  series  certainly  converges  and  its  sum  is  sin  x  for  all  valurs  of  x, 
since  lim  |  £"„  |  =  0. 

Example  2.     Derive  an  infinite  series  for  e*  in  powers  of  (x  —  2). 

Since /(x)  =  c*,  we  have  f'(x)-e',  ••-,  /(")(x)  =  e*  ;  hence  /(2)  =  e^, 
/'(2)  =  e2,  •■•,/'»>(2)  =  e2,  and  If^^^x)  |  ^  e'>  where  b  is  the  largest  value 
of  X  we  shall  considei*.    Then  the  series 


e»  =  e2  +  e2(x-2)  +  ^(x-2)2+  ...  +-^(x-2)»+  ••• 
2  1  n\ 

=  e2[l+(x-2)  +  ±(x-2)2+...+-l(x-2)»4--] 

converges  and  its  sum  is  e',  for  all  values  of  x  less  than  b  ;  for 
lim  \E„\<  lim  «'"]■'''- -]"  =  q. 

Since  b  is  any  number  we  please,  the  series  is  convergent  and  its  sum  is 
e*  for  all  values  of  x. 

EXERCISES  LVm.  — TAYLOR  SERIES 

1.  Obtain  the  infinite  Taylor  series  for  cos  x  in  powers  of  x.     Show 
that  lim  |  ^„  |  =  0. 

2.  Derive   the  following  series,  and  account,  when  possible,  for  the 
fact  that  lim  I  ^„  I  =  0  : 

(c)  e'  =  l  +  x  +  xV2  !  +  xV3  !+•••;  (all  x). 

(b)  e-'  =  l-x  +  x2/2  !  -  xV3  !  +  •••;  (all  x), 

(c)  tan  X  =  X  +  xV3  +  2x5/15  +  17x7315  +  .••  ;  (  |  x  |<  7r/2). 

(d)  log  (1  +  x)  =  X  -  x2/2  +  xV3  -  xV4  +  ••• ;  (  |  x  |<  1). 

(e)  sinh  x  =  (e'  -  e-')/2  =  x  +  xV3  !  +  x^/5  !  +  •••;  (all  x). 
(/)  cosh  X  =  (e^  +  e-')/2  =  1  +  x^l'2  !  +  xV4  !  +  •••;  (all  x). 

(jr)  tanh  X  =sinhx/coshx  =x— xV3  +  2xV15-17xV3l5+ •••;  (all  x). 

3.  Show  that  the  series  of  Ex.  2  (e)  can  be  obtained  from  those  of 
Exs.  2  (a)  and  2  (6)  if  the  terms  are  combined  separately. 

4.  Show  that  the  series  of  Ex.  2  (6)  results  from  the  series  of  Ex.  2  (a) 
if  X  is  replaced  by  —  x. 


276  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  142 

5.  Obtain  the  series  for  sin  x  in  powers  of  (x  —  7r/4). 

6.  Obtain  the  series  for  e^  in  terms  of  powers  of  (x  —  1). 

7.  Obtain  the  series  for  log  x  in  powers  of  (x  —  1).  Compare  it  with 
the  series  of  Ex.  2  (d) . 

8.  Obtain  the  series  for  log  ( 1  —  x)  in  powers  of  x,  directly  ;  also  by 
replacing  x  by  —  x  in  Ex.  2(d). 

9.  Using  the  fact  that  log  [(1+x) /(I -x)]  =  log  (1+x)  -  log  (1-x), 
obtain  the  series  for  log  [(1  +  x)/(l  —  x)]  by  combining  the  separate 
terms  of  the  two  series  of  Ex.  8  and  of  Ex.  2  (d).  This  series  is  actually 
used  for  computing  logarithms. 

10.  Show  that  the  terms  of  the  expansion  of  (a  +  x)"  in  powers  of  x 
are  precisely  those  of  the  usual  binomial  theorem. 

11.  Show  that  the  series  for  e°'+'  in  powers  of  x  is  the  same  as  the 
series  for  e*  all  multiplied  by  e". 

12.  Show  that  the  series  for  10"^  is  the  same  as  the  series  for  e*  with 
X  replaced  by  x/3/,  where  3/ =  2.30  •••. 

143.  Precautions  about  Infinite  Series.  There  are  several 
popular  misconceptions  concerning  infinite  series  which  yield 
to  very  commonplace  arguments. 

(a)  Infinite  series  are  never  used  in  computation.  Contrary  to  a 
popular  belief,  infinite  series  are  never  used  in  computation,  and 
can  never  be  used.  This  is  because  no  one  can  possibly  add  all 
the  terms  of  an  infinite  sei'ies.  What  is  actually  done  is  to  use 
a  few  terms  (that  is,  a  polynomial)  for  actual  computation ;  one 
may  or  may  not  consider  how  much  error  is  made  in  doing  this, 
with  an  obvious  effect  on  the  trustworthiness  of  the  result. 

Thus  we  may  write  ■> 

^3        ^5  r2*+l 

smx=:  X-  —  +  - ± — T  •••  (forever); 

3!      5!  (2^  +  1)!  ^  ^' 

but  in  practical  computation,  we  decide  to  use  a  few  terms,  say  sin  x  =  x 
—  x'y3  !  +x^/5  !.  The  error  in  doing  this  can  be  estimated  by  §  134,  p.  257. 
It  is  I  ^7 1  <|x"/7  1 1 .  For  reasonably  small  values  of  x  [say  |  x  |  <  H'^  <  1/-4 
(radians)],  |  £"7 1  is  exceedingly  small. 

Many  of  the  more  useful  series  are  so  rapid  in  their  convergence  that 
it  is  really  quite  safe  to  use  them  without  estimating  the  error  made  ;  but  if 
one  proceeds  without  any  idea  of  how  much  the  error  amounts  to,  one  usu- 


VIII,  §143]    INFINITE  SERIES  — PRECAUTIONS  277 

ally  computes  more  terms  than  necessary.  Thus  if  it  were  required  to  calcu- 
late sin  14°  to  eight  decimal  places,*most  persons  would  suppose  it  necessary 
to  use  quite  a  few  terms  of  the  preceding  series,  if  they  had  not  estimated  E^. 

(b)  No  faith  can  be  placed  in  the  fact  that  the  terms  are  becom- 
ing smaller.  The  instinctive  feeling  that  if  the  terms  become 
quite  small,  one  can  reasonably  stop  and  suppose  the  error 
small,  is  unfortunately  not  justitied.* 

Thus  the  series       1111  1 

_L  4.  _L  +  _L  +  J- + ...  +  _i_  +  ... 
10      20      30      40  10  ?i 

has  terms  which  become  small  i-ather  rapidly  ;  one  instinctively  feels  that 
if  about  one  hundred  terms  were  computed,  the  rest  would  not  affect  the 
result  very  much,  because  the  next  term  is  .001  and  the  succeeding  ones 
are  still  smaller.     This  expectation  is  violently  wrong. 

As  a  matter  of  fact  this  series  diverges ;  we  can  pass  any  conceivable 
amount  by  continuing  the  term-adding  process.     For 

^  +  T*o  +  -  +  xb  >  8  •  xk  =  irs, 
and  so  on  ;  groups  of  terms  which  total  more  than  1/20  continue  to  appear 
forever  ;  twenty  such  groups  would  total  over  1  ;  200  such  groups  would 
total  over  10 ;  and  so  on.  The  preceding  series  is  therefore  very  decep- 
tive ;  practically  it  is  useless  for  computation,  though  it  might  appear  quite 
promising  to  one  who  still  trusted  the  instinctive  feeling  mentioned  above. 

(c)  If  the  terms  are  alternatehj  ^iositive  and  negative,  and  if  the 
terms  are  numerically  decreasing  loith  zero  as  their  limit,  the  in- 
stinctive feeling  just  mentioned  in  (b)  is  actually  correct :  the 
series  a^  —  Oj  -f  ag  —  ag  -j —  converges  if  a„  approaches  zero ;  the 
error  made  in  stopping  with  a„  is  less  than  a„+i.t 

For,   the   sum   s„  =  Oo  — cfj  +  •••  ±  a„_i   evidently   alternates 

*  This  fallacious  instinctive  feeling  is  doubtless  actually  uxed,  and  it  is  re- 
sponsible for  more  errors  than  any  other  single  fallac-y.  The  example  here 
mentioned  is  certainly  neither  an  unusual  nor  an  artificial  example. 

tOne  must,  however,  make  quite  sure  that  the  terms  actually  approach 
zero,  not  merely  that  they  become  rather  small  ;  the  addition  of  .0000001  to 
each  term  would  often  have  no  appreciable  effect  on  the  appearance  of  the  first 
few  terms,  but  it  would  make  any  convergent  series  diverge. 


278  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  143 

between  an  increase  and  a  decrease  as  n  increases,  and  this  alter, 
nate  swinging  forward  and  then  backward  dies  out  as  n  increases, 
since  a„  is  precisely  the  amount  of  the  nth  swing. 

On  each  swing  s„  passes  a  point  S  which  it  again  repasses  on 
the  return  swing ;  and  its  distance  from  that  point  is  never 
more  tlian  the  next  swing, — never  more  than  a„+i.  Since  a„ 
approaches  zero,  s„  approaches  S,  as  n  becomes  infinite. 

Thus  the  series  for  sin  x  is  particularly  easy  to  use  in  calculation  :  the 
error  made  in  using  x  —  x^/fi  !  in  place  of  sin  x  is  certainly  less  than  x^/b  !. 
The  test  of  §  134  shows,  in  fact,  that  the  error  |  E5\<CM5\x^/b  !|,  where  M5  =  l. 

The  similar  series  for  e'  : 

e»  =  1  +  — +  i5!.4-  ...  +  ^  +  ... 
1  !      2  !  ^       ^  n\ 

is  not  quite  so  convenient,  since  the  swings  are  all  in  one  direction  for 
positive  values  of  x ;  certainly  the  error  in  stopping  with  any  term  is 
greater  than  the  first  term  omitted.  The  error  can  be  estimated  by  §  134, 
p.  257  ;  thus  E^  (for  a;  >  0)  is  less  than  M5  x^/b  !,  where  3/5  is  the  maximum 
of  p{x)  —  e^  between  x  =  0  and  x  =  x,  i.e.  e' ;  hence  E5<e'^x^/h\. 
Note  that  e»  >  1  f  or  x  >  0. 

Another  means  of  convincing  oneself  that  the  preceding  series  converges 
is  by  comparison  with  a  geometric  series  with  a  ratio  x/2,  as  in  Example 
2,  p.  272.  But  this  method  would  require  the  computation  of  a  vast 
number  of  terms,  to  make  sure  that  the  error  is  small. 

(d)  A  consistently  small  error  in  the  values  of  a  function  may 
make  an  enormous  error  in  the  values  of  its  derivative. 

Thus  the  function  y  =  x—  .00001  sin  (100000 x)  is  very  well  approxi- 
mated by  the  single  term  y  =  x,  —  in  fact  the  graphs  drawn  accurately 
on  any  ordinary  scale  will  not  show  the  slightest  trace  of  difference  be- 
tween the  two  curves.  Yet  the  slope  of  y  =  x  is  always  1,  while  the  slope 
oiy  =  x  —  .00001  sin  (100000  x)  varies  from  0  to  2  with  extreme  rapidity. 
Draw  the  curves,  and  find  dy/dx  for  the  given  function. 

One  advantage  in  Taylor  series  and  Taylor  approximating 
polynomials  is  the  known  fact  —  proved  in  advanced  texts  — 
that  differentiation  as  ivell  as  integration  is  quite  reliable  on  any 
valid  Taylor  aiyproximation* 

*  See,  e.g.,  Goursat-Hedrick,  Mathematical  Analysis,  Vol.  I,  p.  380. 


VIII,  §143]    INFINITE  SERIES  — PRECAUTIONS  279 

Thus  an  attempt  to  expand  the  function  y  =  x  —  .00001  sin  (100000  x) 
In  Taylor  form  e;lves 

^  L         a!  5!  J' 

which  would  never  be  mistaken  for  y  —xhy  any  one ;  the  series  indeed 
converges  and  represents  y  for  every  value  of  x,  but  a  very  hasty  exam- 
ination is  sufficient  to  show  that  an  enormous  number  of  terms  would 
have  to  be  taken  to  get  a  reasonable  approximation,  and  no  one  would  try 
to  get  the  derivative  by  differentiating  a  single  term. 

If  the  relation  expressed  by  the  given  equation  was  obtained  by  ex- 
periment, however,  no  reliance  can  be  placed  in  a  formal  differentiation, 
even  though  Taylor  approximations  are  used,  for  minute  experimental 
errors  may  cause  large  errors  in  the  derivative.  Attention  is  called  to  the 
fact  that  the  preceding  example  is  not  an  unnatural  one,  —  precisely  such 
rapid  minute  vibrations  as  it  contains  occur  very  frequently  in  nature. 

EXERCISES  LIX.  — INFINITE   SERIES 

1.  Show  that  the  series  obtained  by  long  division  for  1  -?-  (1  +  x)  is 
the  same  as  that  given  by  Taylor's  Series. 

2.  Obtain  the  series  for  log  (1  -f-  a*)  (see  Ex.  2  (d),  List  LVIII),  by 
integrating  the  terms  of  the  series  found  in  Ex.  1  separately. 

3.  Find  the  first  four  terms  of  the  series  for  sin-i.c  in  powers  of  x  directly ; 
then  also  by  integration  of  the  separate  terms  of  the  series  for  l/Vl— x^. 

4.  Proceed  as  in  Ex.  3  for  the  functions  tan-i  x  and  1/(1  +  x-). 

5.  Show  that  the  series  for  cos  x  in  powers  of  x  is  obtained  by  differ- 
entiating separately  the  terms  of  the  series  for  sin  x. 

6.  Show  that  repeated  differentiation  or  integration  of  the  separate 
terms  of  the  series  for  e^  always  results  in  the  same  series  as  the  original  one, 

7.  From  the  series  for  tan-ix  compute  ir  by  using  the  identity 
ir/4  =  i  tan-1  (1/5)  -  tan-i  (1/239). 

8.  Six)^f;isinu/u)au  =  x-lf^  +  lf-:.. 
Show  that  S(.l)  =  .0999+  ;  8(1)  =  .94G1  ;  ^(3)  =  1.8487. 

9.  The  Gudermannian  of  x  is  gd(x)  =  2  tan-'e'  -  ir/2  ;  expand  in 
powers  of  x  ;  calculate  gd(.l)  =  5°  43',  and  gd(.l)  =  37°  11'. 

10.   The  Fresnel  integrals  are 

C(.)  =-^  f --^d.;  S(z)=-^r'^dz. 
V2irJo     y/z  y/2irJo     y/z 


280  POLYNOMIAL  APPROXIMATIONS    [VIII,  §  143 

Obtain  power  series  in  z  for  C{z)  and  Si^z).  Calculate  C(.l)  =  .2521, 
C(l)  =  .7217,  (7(3)  =  .5610  ;  <S(3)  =  -7117,  ^(.1)  =  .0924,  ^(5)  =  .4659. 

11.  The  graphs  of  Giz)  and  Siz)  (Ex.  10)  are  wave  curves  of  de- 
creasing amplitude.    Locate  the  crests  of  the  waves.    Draw  the  graphs. 

12.  The  "  error  integral  "  is  F{x)  =  — ::  f^e-^^dr,.    Express  P(x)  as  a 

Vtt  Jo 
series  in  powers  of  x  ;  calculate  P(.l)  =  .  1125,  P(l)  =  .8427,  P(2)  =  .9963+. 

13.  Show  that  pVFe-'d«=.8862+.  15.  Show  that  pdi/Vl-«3  =  .508+. 

14.  Show  that  P«<'.3e-«(?i=.8975+.  16.  Showthat  rd«/\/l^=1.311+. 

17.  Show  that  f^^^in  s/*xdx  =  .9309+. 

18.  J5r=  r^'  (1/Vl-A;2sin2  0)  ^0  =  ^  [1  +  (1/2)2A;2  +  (1  •  3/2  •  ^^k^  + 

(1  •  3  •  5/2  •  4  •  6)'-^  A;''  H ].     Obtain  this  result.     The  time  of  swing  of  a 

simple  pendulum  of  length  I  through  an  angle  a  is  ^y/l/g  K,  where  k  = 
sin  (a/2).  Compute  this  time  when  a  =  60°.  (See  Ex.  15,  p.  254  ;  and 
Tables,  V,  D.) 

19.  E  =  Cj^^Vl-k^  sm^<p  d^=  ^[l-(l/2)-^A:2-  [(1  ■  3)/(2-4)]2(A:V3) 

-[(1.3.  5)/(2  •  4  .  6)]2(A;6/5) ].     Obtain  this  result.     Show  that  the 

perimeter  of  an  ellipse,  whose  major  axis  is  2  a  and  eccentricity  k,  is  4  aE. 
Calculate  to  1  %  this  length  when  a  =2 and  k  =  1/2.     (See  Tables,  V,  E.) 

20.  The    Bessel    Function    of    order    zero    is    defined    by    Jo  (.x)  = 

1  -  (x/2)V(l  !)2  +  (x/2)V(2  !)-^  -(x/2)6/(3  \f  +  ..-. 
Calculate  :  Jo(.2)  =  .9900  ;  Jo{\)  =.7652  ;  Jo(S)  =  -  .2601. 
Show  that  Jo{x)  is  a  solution  of  the  equation  y"  +  y'/x  —  y  =  0. 

21.  The    Bessel    Function     of    order    one    is     Ji(x)  =  (x/2)  [1  —  J 
(x/2)V(l  .  2)  +  (x/2)V(l  -2.2.  3)  -(x/2)V(l  •2.3.2.3.4)+  ••.]. 

Calculate  :  ,/i(.2)  =  .0995  ;  ,/i(l)  =  .4401  ;  Ji{S)  =  .3391. 

Show  that  Ji(x)  satisfies  the  equation  y"  +  y'/x  +  (1  +  l/x-)?/  =  0. 

22.  In  the  flow  of  water  through  a  channel  or  pipe  the  "mean 
hydraulic  radius"  is  defined  as  "cross  section  of  stream  -=- wetted  pe- 
rimeter." Calculate  the  m.  h.r.  for  an  elliptical  pipe  flowing  full,  the 
axes  of  the  ellipse  being  4  in.  and  3  in.  respectively.  (See  Ex.  19.) 
Compare  with  result  for  a  circular  pipe  of  the  same  cross  section. 


i 


CHAPTER   IX 

SEVERAL  VARIABLES      PARTIAL   DERIVATIVES 
APPLICATIONS      GEOMETRY 

PART   I.     PARTIAL    DIFFERENTIATION— ELEMENTARY 
APPLICATIONS 

144.  Partial  Derivatives.  If  one  quantity  depends  upon 
two  01-  more  other  quantities,  its  rate  of  change  with  respect 
to  one  of  them,  while  all  the  rest  remain  fixed,  is  called  a 
partial  derivative.* 

If  z  =/(;f,  y)  is  a  function  of  x  and  y,  then,  for  a  constant 
value  of  y,  y='k,z  is  a  function  of  x  alone :  z  =f(x,  k)  ;  the 
derivative  of  this  function  of  x  alone  is  called  the  partial 
derivative  of  z  with  respect  to  ^,  and  is  denoted  by  any  one 
of  the  symbols 

dz  _dfioc,y)       f  ,        s  _(lf(ocJc) 

=  lim  f^^  "*"  ^'^'  ^^  ~  ^'^'^'  ^^ 

A  precisely  similar  formula  defines  the  partial  derivative 
of  z  with  respect  to  y  which  is  denoted  by  dz/dy. 

In  general,  if  w  is  a  function  of  any  number  of  variables  x, 
y,  z,  •■•  ,  and  if  one  calculates  the  first  derivative  of  u  with 

*  This  notion  is  perhap.s  more  prevalent  in  the  world  at  large  than  the 
notion  of  a  derivative  of  a  function  of  one  variable,  becau.se  quantities  in 
nature  usually  depend  upon  a  great  many  influences.  The  notion  of  parthtl 
dericative  is  what  is  expressed  in  the  ordinary  phrases  "the  rate  at  which  a 
quantity  changes,  everything  else  being  supposed  equal,"  or  "...  other 
things  being  the  same."  The  reason  for  the  existence  of  this  idea  is  the 
attempt  to  estimate  the  effect  of  each  contributing  cause  apart  from  that  of 
all  others.  Compare  Ex.  15,  p.  254,  Ex.  24,  p.  90,  and  many  others. 
281 


282  SEVERAL   VARIABLES  [IX,  §  144 

respect  to  each  of  these  variables,  supposing  all  the  others  to 
be  fixed,  the  results  are  called  the  first  partial  derivatives  of 
M  with  respect  to  x,  y,  z,  •  •  •,  respectively,  and  are  denoted  by 
the  symbols 

du/dx,    du/dy,    du/dz,  •••  . 

145.    Technique.     No  new  rules  are  necessary. 

Exarnple  1.     Given  z  =  af-\-y',  to  find  dz/dx  and  dz/dy. 

To  find  dz/dx,  think  of  y  as  constant :  y  =  k]  then 

dz^d(^±f)^d(c^+]f)^^^,    dz^2y. 
dx  dx  dx  '    dy 

Example  2.       Given  z  =  a^  sin  (x  +  y-) ,  to   find   dz/dx  and 
dz/dy. 

dz  _  d]o(^sm(x+y-)\  _  Fd^x^  sin 


dx  dx  1_  dx 

=  2  a;  sin  (a;  +  y^)  +  x^  cos  (x  +  y^). 

dz  ^d\a^sm(x  +  y^)\  ^  fd Ik'  sin  (fc  +  y") H 
dy  dy  |_  dy  J^ 

=  2  afy  cos  (x  +  y'). 

146.  Higher  Partial  Derivatives.  Successive  differentiation 
is  carried  out  as  in  the  case  of  ordinary  differentiation. 
There  are  evidently  four  ways  of  getting  a  second  partial 
derivative :  differentiating  twice  with  respect  to  x ;  once  with 
respect  to  x,  and  then  once  with  respect  to  y;  once  with 
respect  to  y,  and  then  once  with  respect  to  x ;  twice  with  re- 
spect to  y.  These  four  second  derivatives  are  denoted,  re« 
spectively,  by  the  symbols 


IX,  §  146J  PARTIAL   DERIVATIVES  283 

There  is  no  new  difficulty  in  carrying  out  these  operations ; 
in  fact  the  situation  is  simpler  than  one  might  suppose,  for 
it  turns  out  that  the  two  cross  derivatives  f^^  and  /^^  are 
always  equal;  the  order  of  differentiation  is  immaterial.* 

A  similar  notation  is  used  for  still  higher  derivatives : 

•''"     dx^     fxUx-j     ""    dycx^     dyW)' 
etc. ;  and  the  order  of  differentiation  is  immaterial. 

The  order  of  a  partial  derivative  is  the  total  number  of  successive  dif- 
ferentiations performed  to  obtain  it.  The  partial  derivatives  of  the  first 
and  second  orders  are  very  frequently  represented  by  the  letters 
P,  q,  r,  s,  t : 

^  ~  gx'  ^  "~  fy '  *"  ~  ex2'  *  ~  dxdy  ~~  dycx '     ~  dy-' 

Example  1.     Given  z  =  x-  sin  (x  +  y-),  show  that/^j,=/„j. 

Continuing  Example  2,  §  145,  we  find : 

i!L  ^  i  f  ^\  =  1  [2  a;  sin  (x  +  y2)  +  x2  cos  (x  +  y^)! 
dycx     dy\dxj     dyL  J 

=  ixy  cos  (x  +  j/2)  _  2  x-y  sin  (x  +  y^). 

i^  =1  f^U  A  [2  X2y  cos  (X  +  2/2)-| 

dxdy     dx\dyj     dxL  J 

=  4  xy  cos  (x  +  y-)  —  2  x-y  sin  (x  +  y"^). 

EXERCISES  LX.  —  TECHNIQUE  OF  PARTIAL   DIFFERENTIATION 

,  1.  The  volume  of  a  right  circular  cylinder  is  v  =  irr-h.  Find  the  rate 
of  change  of  the  volume  with  respect  to  r  when  h  is  constant,  and  ex- 
press it  as  a  partial  derivative.     Find  dv/dh,  and  express  its  meaning. 

2.  The  pressure  p,  the  volume  v,  and  temperature  tf  of  a  gas  are  con- 
nected by  the  relation  pv  —  kd,  where  6  is  measured  from  the  absolute 
zero,  —  273°  C.  Assuming  0  constant,  find  dp/cv  and  express  its  mean- 
ing. If  the  volume  is  constant,  express  the  rate  of  change  of  pressure 
with  respect  to  the  temperature  as  a  derivative,  and  find  its  value. 

*  .\t  least  if  the  derivatives  are  themselves  continuous.  See  Goursat- 
Hedrick,  Mathematical  Analysis,  Vol.  I,  p.  13. 


284  SEVERAL  VARIABLES  [IX,  §  147 

3.  Find  the  rate  of  change  of  the  volume  of  a  cone  with  respect  to  its 
height,  if  the  radius  is  constant ;  and  the  rate  of  change  of  the  volume 
with  respect  to  the  radius,  if  the  height  is  a  constant. 

4.  Find  the  first  and  second  partial  derivatives,  dz/dx,  dz/dy,  d'^z/dx'^, 
d'^z/dx  cy,  d-z/dy  dx,  and  d'^z/dy'^  for  each  of  the  following  functions.  In 
each  case  verify  the  fact  that  d-z/dx  dy  =  d'^z/dy  dx. 

(a)  z  -x^  —  y-.  (d)   z  =  e-^'+^y.  (g)  z  =  (x  +?/)e-^'+i''. 

(6)  z  =  x^  +  X2/2.  (e)    z  =  tan-i(2//x).  (h)  z  =  (a;y-2 y^)3/2, 

(c)   s=:siu(x2  +  2/2).  (/)  2  =  e- sin  y.  (i)    z^log^x^+y^y/^. 

5 .  Verify  the  fact  that  z  =  x^  —  y^  satisfies  the  equation  d^z/dx^  + 
d^z/dy^  =  0.     Show  that  the  same  equation  is  satisfied  by  4  (e)  and  4  (i). 

[Note.  An  equation  which  contains  partial  derivatives  is  called  a 
partial  differential  equation.  (See  p.  382.)  The  particular  equation 
of  this  exercise  is  called  Laplace's  equation.] 

6.  A  point  moves  parallel  to  the  a--axis.  What  are  the  rates  of 
change  of  its  polar  coordinates  with  respect  to  x  ? 

7.  Show  that  the  rate  of  change  of  the  total  surface  of  a  right  circular 
cylinder  with  respect  to  its  altitude  is  dA/dh  =  2  7rr ;  and  that  its  rate 
of  change  with  respect  to  its  radius  is  dA/dr  =  2  ttA  +  4  Trr. 

8.  Calculate  the  rate  of  change  of  the  hypotenuse  of  a  right  triangle 
relative  to  a  side,  the  other  side  being  fixed ;  relative  to  an  angle,  the 
opposite  side  being  fixed. 

9.  Two  sides  and  the  included  angle  of  a  parallelogram  are  a,  b,  C, 
respectively.  Find  the  rate  of  change  of  the  area  with  respect  to  each 
of  them,  the  other  two  being  fixed ;  the  same  for  the  diagonal  opposite 
to  C. 

10.  In  a  steady  electric  current  C=:  F-=-  JB,  where  O,  V,  R,  denote 
the  current,  the  voltage  (electric  pressure),  and  the  resistance,  respec- 
tively.    Find  dC/dV  and  dC/dE,  and  express  the  meaning  of  each  of 

them. 

147.  Geometric  Interpretation.  The  first  partial  derivatives 
of  a  function  of  two  independent  variables 

can  be  interpreted  geometrically  in   a  simple  manner.     This 
equation  represents  a  surface,  which  may  be  plotted  by  erect- 


IX,  §  148] 


PARTIAL  DERIVATIVES 


285 


ing  at  each  point  of  the  a^-plane  a  perpendicular  of  length 
f(x,  y) ;  the  upper  ends  *  of  these  perpendiculars  are  the  points 
of  the  surface. 

Let  ABCD  be  a  portion  of 
this  surface  lying  above  an  area 
ahcd  of  the  .x-?/-plane.  If  x 
varies  while  y  remains  fixed, 
say  equal  to  k,  there  is  traced 
on  the  surface  the  curve  HK, 
the  section  of  the  surface  by 
the  plane  y  =  k.  The  slope  of 
this  curve  is  dz/dx. 

Similarly,  dz/dy  is  the  slope 
of  the  curve  cut  from  the  sur-     "  Fig.  63 

face  by  a  plane  x  =  h. 

148.  Total  Derivative.  If  in  addition  to  the  function 
Zz=f{x,  y),  a  relation  between  x  and  y,  say  y=  <}>{x),  is  given, 
z  reduces  by  simple  substitution  to  a  function  of  one  variable  : 

z=f(x,  y),  y  =  <t>(^)    gives   z  =f(x,  <}>(x)). 

Now  any  change  Aar  in  x  forces  a  change  Ay  in  y;  hence  y 
cannot  remain  constant  (unless,  indeed,  (f>(x)  =  const.).  Hence 
the  change  Az  in  the  value  of  z  is  due  both  to  the  direct 
change  Ax  in  x  and  also  to  the  forced  change  Ay  in  y.  We 
i^shall  call 

Az=  the  total  change  in  z  =  f(x+  Ax,  y  +  Ay)  —f(x,  y), 
A^z=  the  partial  change  due  to  Ax  directly 

=  f{x+Ax,y)-f(x,y), 
A/=the  partial  change /orced  by  the  forced  change  Ay 
=  Az-A^,  =f{x  +  Ax,  y  +  Ay)  -fix  +  Ax,  y). 

•  If  z  is  negative,  of  course  the  lower  end  is  the  one  to  take. 


286 


SEVERAL  VARIABLES 


[IX,  §  148 


It  follows  that 


,-,s      dz      T     Az 
(1)     —  =  lim  — 

dx       Ax=y)  Ax 


{' 


^)}' 


^lim^-'  +  V 

ajs£=o       Aa; 


I       Ay^   \  AX 


I  /•(y  +  Aa-.  y  +  Ay)  -  /  («•  +  Aa-.  y)  Ay  N  1 
Ay  ^"^  /  j ' 

whence,  if  the  partial  derivatives  exist  and  are  continuous,* 


(2) 


dz 
dx 


Ax=oL  Ax        Ay  Ax  J 


dz   .   dz  dy 
dx      dydx* 


or,  multiplying  both  sides  by  dx{=Ax) 
(3) 


dz  =  ^dx-\-^ dy,  since  dy  =  ^ dx, 

Sy  dx 


dx 


v^               Qo 

^  V=0(a:) 

Fig.  64 

PS  =  AB  =  Ax 

ST=  C2)  =  Ay 

SR  =  A«3  =  TM=  R„R  -  PgP 

MQ  =  A^»lx=^+Ax  =  QoQ  -  ^off 

A  3  =  ^0^  -  PoP=  TQ^SS+MQ 

=  V+\^].=. 

+Ai 

where  %  =  ^'(.^')  dx.  Since  <^(a;)  is 
any  function  whatever,  dy  is  really 
perfectly  arbitrary.  Hence  (3)  holds 
for  any  arbitrary  values  of  dx  and  dy 
whatever,  where  dz=(dz/dx)  dx  is 
defined  by  (2) ;  dz  is  called  the  total 
differential  of  z. 

These  quantities  are  all  repre-^ 
sented  in  the  figure  geometrically: 
thus  Az  =  A^z  +  AyZ  is  represented  by 
the  geometrical  equation  TQ  =  SR 
+MQ.  It  should  be  noticed  that  dt 
is  the  height  of  the  plane  drawn 
tangent  to  the   surface   at  P,  since 


*  For  a  more  detailed  proof  using  the  law  of  the  mean,  see  Goursat-Hed* 
rick,  Mathematical  Analysis,  I,  pp.  38-42. 


IX,  §  149]  PARTIAL   DERIVATIVES  287 

dz/dx  and  dz/dy  are  the  slopes  of  the  sections  of  the  surface  by 
y  =yp  and  x=  Xp,  respectively.     [See  also  §  1G4,  p.  321.] 

If   the   curve   PqQq  in  the  a;j/-plane  is  given   in  parameter 
form,  x=<f>(t),  y  =  i}/(t),  we  may  divide  both  sides  of  (3)  by  dt 
and  write 
/ . .  dz  _  dz  dx      dz  dy 

^  ^  Tt~^Yt    TyW 

since  dx  h-  dt  =  dx/dt,  dy  -=-  dt  =  dy/dt. 

149.  Elementary  Use.  In  elementary  cases,  many  of  which 
have  been  dealt  Avith  successfully  before  §  148,  the  use  of  the 
formulas  (2),  (3),  and  (4)  of  §  148  is  quite  self-evident. 

Example  1.     The  area  of  a  cylindrical  cup  with  no  top  is 
(1)  ^  =  2  7rr/i  +  7rr2, 

where  h  is  the  height,  and  r  is  the  radius  of  the  base.  If  the  volume  of 
the  cup,  irr-h,  is  known  in  advance,  say  irr^A  —  10  (cubic  inches),  we  ac- 
tually do  know  a  relation  between  h  and  r  : 

whence 

(3)  ^:^2  7rr^+7rr2=^  +  7rr2 

irr-  r 

from  which  dA/dr  can  be  found.     We  did  precisely  the  same  work  in 

Ex.  7,  p.  68.     In  fact  even  then  we  might  have  used  (1)  instead  of  (3), 

ind  we  might  have  written 

U)     ^  =  2  irr—  -I-  2  tt/H-  2  Trr,  or  dA  =  2  nr  dh  +  (2  irh  +  2  irr)dr, 
dr  dr 

where  dh/dr  is  to  be  found  from  (2). 

This  is  precisely  what  formula  (2),  §  148,  does  for  us  ;  for 

(5)     ?ii  =  2  7rft  +  2,rr,    ^  =  27rr, 

—  =(2  7r;i  +  2irr)+(2  7r?-)— ,  or  dA  =(2  7rh  +  2irr)dr  +  2  rrrdh. 
dr  dr 

We  used  just  such  equations  as  (4)  to  get  the  critical  values  in  finding 
extremes  for  dA/dr  =  0  at  a  critical  point.  We  may  now  use  (2),  §  148, 
to  find  dA/dr ;  and  the  work  is  considerably  shortened  in  some  cases. 


288  SEVERAL  VARIABLES  [IX,  §  149 

Example  2.  The  derivative  dy/dx  can  be  found  from  (2),  §  148,  if  we 
know  that  z  is  constant. 

Thus  in  §  26,  p.  44,  we  had  the  equation 

(1)  a;2  +  2/2  =  1, 
and  we  wrote : 

(2)  ^(^i±l!i  =  2x  +  22/^  =  ^ffi  =  0, 

dx  dx       dx 

whence  we  found 

(3)  x  +  ,f^=:0,  or^  =  -5. 

dx  dx         y 

This  work  may  be  thought  of  as  follows  : 

Let  0  =  a;2  +  j^  ;  then 

dz  ^d{x''  +  y^)^dz  ^  ^zdy^^^  I  2^*^^; 
dx  dx  dx     dy  dx  dx' 

but ;?  =  1  by  (1)  above  ;  hence  dz/dx  =  0,  and 

2x  +  2y^  =  0,  or  ^  =  -^. 
dx  dx         y 

Thus  the  use  of  the  formulas  of  §  148  is  essentially  not  at  all 
new ;  the  preceding  exercises  and  the  work  we  have  done  in 
§§  26,  34,  etc.,  really  employ  the  same  principle.  But  the 
same  facts  appear  in  a  new  light  by  means  of  §  148 ;  and  the 
new  formulas  are  a  real  assistance  in  many  examples. 

150.  Small  Errors.  Partial  Differentials.  Another  applica- 
tion closely  allied  to  the  work  of  §  132,  p.  252,  is  found  in  the 
estimation  of  small  errors. 

Example  1.  The  angle  vl  of  a  right  triangle  ABG  (C  =  90^),  may  be 
computed  by  the  formula 

tan^=^,  or  ^  =  tan-i  ?, 

where  a,  5,  c  are  the  sides  opposite  A,  B,  C.  If  an  error  is  made  in 
measuring  a  or  b,  the  computed  value  of  A  is  of  course  false.  We  may 
estimate  the  error  in  A  caused  by  an  error  in  measuring  a,  supposing  tem- 
porarily that  b  is  correct,  by  §  132 ;  this  gives  approximately 


IX,  §  150]  PARTIAL   DERIVATIVES  289 

1 

A.A  =  ^Aa  =  —^Aa  =  — ^  Aa, 
da  1  +  ^  «'"  +  b- 

where  d  is  used  in  place  of  cZ  of  §  132,  since  ^1  really  depends  on   b 
also,  and  we  have  simply  supposed  b  constant  temporarily.     Likewise  the 
error  in  A  caused  by  an  error  in  b  is  approximately, 
_  a 

AU  =  Ma6  =  —^,Ab  =  -F^A6. 
cb  I  4-11  «■"  +  b'^ 

If  errors  are  possible  in  both  measurements,  the  total  error  in  A  is, 
approximately,  the  sum  of  these  two  partial  errors  : 

\AA\^\A^A\  +  \A,A\=^-^^^1+^^1^^- 
a'^  +  b- 

The  methods  of  §  133,  p.  256,  give  a  means  of  finding  how  nearly  cor- 
rect these  estimates  of  AaA,  A^A,  and  A.-l  are  ;  in  practice,  such  values 
as  those  just  found  serve  as  a  guide,  since  it  is  usually  desired  only  to 
give  a  general  idea  of  the  amounts  of  such  errors. 

This  method  is  perfectly  general.  The  differences  in  the 
value  of  a  function  z  =f{x,  y)  of  two  variables,  x  and  y,  which 
are  caused  by  differences  in  the  value  of  x  alone,  or  of  y  alone, 
are  denoted  by  ^^z,  ^^z,  respectively.  The  total  difference  in  z 
caused  by  a  change  in  both  x  and  y  is 
Az  =f(x  +  Ax,  y  +  Ay) -fix,  y) 

=  lf{x  +  Ax,y-\-Ay)-f{x  +  Ax,y)-]-\-lf{x+Ax,y)-f{x,y)-] 

as  in  §  148.     The  differences  A^z  and  A^z  are,  approximately,* 


-1 


*  More  precisely,  these  errors  are 

A:,Z  =  ^    .    AZ  +  £'2,      A,Z  =  ^   I  •  A?/  +  E' 

where  |  E'^  \  and  1  £"2 1  are  less  than  the  maximum  M^  of  the  values  of  all  of 
the  second  derivatives  of  2  near  (r,  ?/)  multiplied  by  Ax^,  or  Ay^,  respectively 
(see  §133).     And  since  dz/dy  is  itself  supposed  to  be  continuous,  we  may 

AZ  =  —  AX  +   —  A?/  +  T'-i, 
?j-  ru 

where  \  E-i\  is  less  than  3/2(  I  Az  I  +  I  A^  I  )2.  [Law  of  the  Mean.  Compare 
§133.] 

V 


290  SEVERAL  VARIABLES  [IX,  §  150 


A^z  =  ■ —  Act-,    A„2!  =  —  i 
dx  dy 


whence,  approximately, 


Az  =  Az  +  A^z  =  —Ax-  +  —  Ay. 
ax  dy 

The  products  {dz/dx)dx  and  {dz/dy)dy  are  often  called  the 
partial  differentials  of  z,  and  are  denoted  by 

d^z  =  —-dx,  d  z  =  — dy,  whence  dz  =  d^z  +  d  z, 
ox  dy 

where  dx  =  Ax  and  dy  ={dy/dx)Ax=  Ay,  approximately.     We 
have  therefore,  approximately, 

Az  =  d^z  +  dyZ, 

within  an  amount  which  can  be  estimated  as  in  §  133  and  in 
the  preceding  footnote. 

Similar  formulas  give  an  estimate  of  the  values  of  the  changes  in  a 
function  ri  =f(x,  y,  z)  of  the  variables  ,r,  y,  z\  we  have,  approximately, 

A^?f  =  —Ax,  ^yU  =  —Ay,    A,u  =  —Az, 
dx  dy  dz 

Am  =  A^M  +  AyU  +  A,u  =  —Ax  +  —Ay  +  —Az, 
dx  dy  dz 

within  an  amount  vphich  can  be  estimated  as  in  the  preceding  footnote. 
The  generalization  to  the  case  of  more  than  three  variables  is  obvious. 


EXERCISES  LXI.  — TOTAL  DERIVATIVES  AND  DIFFERENTIALS 

1.  Express  the  total  surface  area  A  of  a  cylindrical  can  vyith  a  bottom 
but  no  top,  in  terms  of  the  height  h  and  the  radius  of  the  base  r.  If  the 
volume  of  the  can  is  given,  say  100  cu.  in.,  find  a  relation  between  h  and 
r  ;  and  find  dA/dr. 

2.  Find  the  most  economical  dimensions  for  the  can  described  in  Ex,  1. 

3.  Find  the  most  economical  dimensions  for  a  funnel  made  in  the  form 
of  a  right  cone,  neglecting  the  outlet  hole. 


IX,  §  150]  PARTIAL   DERIVATIVES  291 

4.  The  pressure  p,  the  volume  v,  and  tlie  temperature  6  of  any  gas  are 
connected  by  the  relation  pv  =  kd,  when  k  is  a  constant.  When  no  heat 
escapes  or  enters  it  is  found  by  experiment  that;^  =  c  •  v-^-^i  for  air.  Ex- 
press 6  in  terms  of  v  alone  and  find  dd/dv.  Find  the  same  result  directly 
by  §  149. 

5.  Find  dz  when  z  is  given  in  terms  of  x  and  y,  and  y  is  given  in 
terms  of  x,  by  one  of  the  following  sets  of  equations  : 


(a)  z  =  x-^  +  y^     2/  =  2x  +  3. 

(d)    z  =  xy,     y  =  V2x  +  3. 

(6)   z  =  x-  -  2/2,     y  =  x3/2. 

(e)     z  =  sin  (r  +  y),     y  =  x. 

(c)    z-x^  —  y"-,     y  =  z. 

(/)  z=Vx^  +  y\    y  =  l/x. 

6.  Find  dz/dt  when  z  =  xy,  and  x  =  sin  f,  y  =  cos  t  by  expressing  2  in 
terms  of  t ;  without  expressing  z  in  terms  of  t.  Interpret  this  result 
geometrically. 

7.  Find  dy/dx  in  each  of  the  following  implicit  equations  by  method 
of  Ex.  2,  §  149  : 

(a)  x2  +  4  y2  =  1.  (c)    x3  +  2/3  _  3  .r2/  =  0. 

(6)  4x2-92/2  =  36.  (rZ)    2/-(2  a  -  x)  =  x'. 

8.  li  A,  B,  C  denote  the  angles,  and  a,  b,  c  the  sides  opposite  them, 
respectively,  in  a  plane  triangle,  and  if  a,  A,  B  are  known  by  measure- 
ments, ft  =  ffl  sin  B/sin  A.  Show  that  the  error  in  the  computed  value 
of  6  due  to  an  error  da  in  measuring  a  is,  approximately, 

dab  =  sin  B  esc  A  da. 
Likewise  show  that 

?Aft  =  —  a  sin  B  CSC  A  ctn  A  dA,  and  csb  =  a  cos  B  esc  A  dB  ; 
and  the  total  error  is,  approximately,  db  =  dab  +  d^b  +  dsb.     Note  that 
A  and  B  are  expressed  in  radian  measure. 

9.  The  measured  parts  of  a  triangle  and  their  probable  errors  are 

a  =  100  ±  .01  ft.,     A  =  100°  ±1',     i?  =  40'^  ±  1'. 
Show  that  the  partial  errors  in  the  side  ft  are 

dab  =  ±  .007  ft.,       dib  =  ±  .003  ft.,       deb  =  ±  .023  ft. 
If  these  should  all  combine  with  like  signs,  the  maximum   total  error 
would  be  db=±  .033  ft. 

10.    If  a  =  100  ft.,  B  :=  40°,  A  -  100°,  and  each  is  subject  to  an  error 
of  1  %,  find  the  per  cent  of  error  in  6. 


292  SEVERAL  VARIABLES  [IX,  §  150 

11.  Find  the  partial  and  total  errors  in  angle  B,  when 

a  =  100  ±  .01  ft.,     6  =  159  ±  .01  ft.,     4  =  30^  ±  1'. 

12.  The  radius  of  the  base  and  the  altitude  of  a  right  circular  cone  being 
measured  to  1%,  what  is  the  possible  percent  of  error  in  the  volume? 
Ans.  3%. 

13.  The  formula  for  index  of  refraction  is  m  =  sin  i/sin  r,  i  being  the 
angle  of  incidence  and  r  the  angle  of  refraction.  If  i  =  50"  and  r  =  40°, 
each  subject  to  an  error  of  1%,  what  is  m,  and  what  its  actual  and  its 
percentage  error  ? 

14.  Water  is  flowing  through  a  pipe  of  length  L  ft.,  and  diameter 
Bit.,  under  a  head  of  ^ft.     The  flow,   in  cubic  feet  per  minute,  is 

Q  =  2356  J    ^^^ —     liL  =  1000,  Z>  =  2,  and  fi-  =  100,  determine  the 

change  in  Q  due  to  an  increase  of  1%  in  ^;  in  i;  in  D.  Compare  the 
partial  differentials  with  the  partial  increments. 

15.  If  the  coordinates  (x,  y)  are  changed  to  polar  coordinates  (p,  ^), 
find  z  in  terms  of  p  and  ^  if  2  =  4  x^  +  y-  ;  find  dz/dp  and  dz/dd. 

16.  Find  dz/dp  and  dz/dd  it  z  =  xy  -  ^  y"^. 

17.  Find  dz/dx  and  dz/dy  if  2  =  p2  —  2  p  cos  6,  where  (p,  6)  are  the  polar 
coordinates  of  the  point  (x,  y) . 

18.  Find  dz/dd  if  ;s  =  x^  —  4  ?/2,  where  x  =  a  tan  6,  y  =  a  sec  0,  by  ex- 
pressing z  in  terms  of  6  ;  without  expressing  z  in  terms  of  d. 

19.  Find  dz/dt  ifz-  e^'+J''  sin  (x^+y) ,  where  x  =  l  +  2t  +  t-,y  =  te-*. 

151.   Significance  of  Partial  and  Total  Derivatives.      The   <. 

formulas  of  §  148  become  of  vital  importance  in  scientific  and 
mathematical  problems.  The  methods  employed  are  illustrated 
by  the  following  typical  examples. 

Example  1.  Expansion  of  a  Gas  at  Constant  Temperature.*  Thus  in 
the  case  of  a  gas  under  pressure  p,  we  have 

(1)  pv  =  ke, 

*  Often  called  isothermal  expansion. 


IX,  §  151]  PARTIAL   DERIVATIVES  293 

where  v  is  the  volume   and   d    is    the   absolute  temperature,    that  is, 
e  —  C  +  273'^  where  C  is  the  temperature  (C).     In  general,  we  have 
f9\  dB  _dB      d6  dp  _p  .  v  dp 

dv      dv      dp  dv      k      k  dv 
If  the  temperature  is  constant  during  a  change  in  volume,  the  pressure 
must  change,  for  dd/dv  =  0,  and  therefore 

(3)  ^(JP ^^  =  0,  or  ^  =  _^^Z^  =  _£, 

dp  dv     dv        '         dv  dd/dp  v 

where  30/ dv  =  p/k  is  the  rate  of  change  of  temperature  which  would 
occur  if  V  alone  were  changed. 

Here  again  the  bare  fact  that  dp/dv  =  —  p/v  can  be  obtained  with- 
out using  §  148.  For  since  ^  is  constant,  pt)= const.,  hence  pdv  +  i;  dp  =  0 
and  dp/dv  =  -  p/v.     (See  Ex.  26,  p.  90.) 

The  new  fact  discovered  by  §  148  is  the  second  equation  in  (3),  which 
says  that  the  rate  of  change  of  pressure  with  respect  to  volume  in  expan- 
sion at  constant  temperature,  is  equal  to  the  negative  of  the  ratio  of  the 
rate  of  change  of  temperature  when  the  volume  alone  changes  to  the  rate 
of  change  of  temperature  when  the  pressure  alone  changes.  This  fact 
remains  strictly  true  even  when  (1)  is  not  strictly  true  ;  for  if  the  tem- 
perature can  be  expressed  as  any  function  of  the  volume  and  the  pressure, 
the  first  equation  under  (3)  remains  true.     (See  Ex.  28,  p.  57.) 

Example  2.  Expansion  when  no  Heat  escapes  or  enters*  The  funda- 
mental equation  j)v  =  kd  of  Ex.  1  and  equation  (2)  hold  true  in  any 
case  for  periect  gases.  If  no  heat  escapes  from  nor  enters  the  gas,  its 
temperature  is  bound  to  rise  or  fall  if  the  product  pv  of  the  pressure  and 
the  volume  does  not  remain  constant ;  for  dd/dv  =  0  if  and  only  if 
dp/dv  =  —  p/v. 

For  any  gas  the  application  of  sudden  mechanical  pressure  —  such  as 
that  of  the  piston  of  an  air  compressor  —  results  in  some  actual  decrease 
in  volume,  but  the  rate  dp/dv  depends  on  the  nature  of  the  particular 
gas.  For  air,  it  is  found  experimentally  that  pv^*^  =  const,  (nearly)  ; 
whence,  from  (1), 

e^m^'^jn^,     ^lP^-lA\c.v-^-»,      ^  =  -OAl'-v-^*K 
k  k  dv  dv  k 

Using  (2),  we  might  have  written 

de^p      v<lp^  ctri;«      V         J  4j  ^^.,«.  ^  _  0  4j  c  ^.,.« 
dv     k     kdv         k         k^  ^  k 


*  Often  called  adiabatic  expansion. 


294 


SEVERAL  VARIABLES 


[IX,  §  151 


These  two  results  for  dd/dv,  found  from  (1)  and  from  (2)  of  Ex.  1,  must 
of  course  agree. 

Example  Z.  Implicit  Equations.  Cuntour  Lines  on  a  Surface.  If  the 
equation  of  a  plane  curve  is  given  in  implicit  form, 

(1)  /(^,2/)=0, 

we  may  think  of  the  curve  as  a  section  of  the  surface 

(2)  z  =Ax,  y) 

by  the  plane  2  =  0  (compare  Ex.  2,  p.  288).     Then  (2),  §  148,  becomes 

(3)  dz_dzdz^dy_Q 
dx      dx      dy  dx 

since  dz/dx  =  dO/dz  =  0.     Hence,  solving  for  dy/dx, 

^  ^  dx         dz/dy' 

The  same  equation  holds  for  any  section  of  the  surface  by  any  horizon- 
tal plane  z  =  k.     Such  a  section  is  often  called  a  contour  line. 

Notice  that  the  value  of  dy/dx  given  by  (4)  is  equivalent  to  the  value 
found  by  the  method  of  §  26,  p.  44.  In  that  paragraph,  a  value  of  dy/dx 
"was  discarded  if  the  point  (3;,  y')  did  not  lie  on  the  given  curve  (1).  It  is 
easy  to  see  now  that  in  any  case  (4)  expresses  the  value  of  dy/dx  at  any 
point  (*,  2/)  for  the  particular  contour  line  through  that  point. 

Example  4.     Floio  of  Heat  in  a  Metal  Plate.     Directional  Derivative. 

Let  us  suppose  that  a  metal  plate  is  steadily  warmed  on  one  edge  {e.g.  by 

a  gas  burner)  and  steadily 
cooled  on  the  other  {e.g.  by 
a  water  jacket).  Then,  after 
a  lapse  of  time,  the  tempera- 
ture at  every  point  in  the 
plate  is  quite  fixed,  though 
the  temperature  is  different 
at  different  points.  Thus, 
the  temperature  0  at  any 
point  (.X,  y)  is  a  function  of 
^^^-  ^^  X  and  of  y, 

(1)  e  =f{x,  y'). 

Let  y  =  (f>  {x)  be  any  curve  through  a  point  P ;  if  we  follow  the  variations 

in  temperature  along  that  curve, 

(2)  e=f{x,y),y  =  <l>{x), 


^cdh 


-op 


IX,  §  151]  PARTIAL  DERIVATIVES  295 

we  have,  by  (3),  §  148, 

(3)  de  =  ^dx  +  ^  dy. 

^  '  ex  cy 

Let  s  be  the  length  of  the  arc  of  the  curve  ;  then  ds"^  =  dx^  +  dy^, 
dx/ds  =  cos  a,  dy/ds  =  sin  a,  where  a-AxDT.hj  §  62,  p.  107  ;  hence, 
dividing  both  sides  of  (3)  by  ds,  we  have 

^  =  L^^  +  e^^  =  £?cos«  +  i^sin«. 
^  ^  ds      dx  ds      dy  ds       ex  dy 

This  equation  shows  that  the  rate  of  change  of  the  temperature  along 
the  curve  depends  only  on  the  angle  a  ;  all  curves  tangent  to  DPT  at  P 
give  the  same  result  for  dd/ds. 

The  equation  (4)  evidently  holds  for  any  function  e=f(x,  y)  what- 
ever ;  often  the  derivative  d9/ds  is  called  a  directional  derivative,  that  is, 
a  derivative  (or  rate  of  change)  of  0  in  the  direction  FT. 

Example  5.  Flow  of  Water  in  Fipes.  Farticle  Derivative.  When 
water  is  flowing  in  a  pipe,  the  speed  of  the  water  may  be  considered  as 
follows :  . 

(a)  Fixing   our   attention  ^ ;2^  A^ 

upon  a  particular  point  ^  ni      y        |  . ^ Jll \ 

the  pipe, — say  its  mouth, —       ( — 


we  may  consider  the  speed  pj^,  gg 

of    various    water    particles 

which  pass  that  point.     This  speed  Sa  may  change  as  time  goes  on,  it 

the  water  pressure  varies  from  any  cause. 

(6)  Fixing  our  attention  on  a  particular  tcater  particle  P,  as  it  moves 
through  the  pipe,  that  particular  particle  has  a  speed,  Sp,  which  may 
change  even  when  the  flow  through  the  pipe  is  perfectly  constant ;  for  if 
F  moves  from  a  wide  part  of  the  pipe  to  a  comparatively  narrow  part  (as 
in  the  nozzle  of  a  hose)  the  speed  Sp  increases.  It  is  clear  that  Sp  =  Sa, 
when  F  is  at  A. 

(c)  Let  us  suppose  the  pressure  is  constant.  Then  Sa  depends  only  on 
the  (fixed)  position  of  the  point  A ;  but  Sp  depends  upon  the  time  t, 
since  the  position  of  F  changes  with  the  time  : 

ct  ct 

{d)  If  the  xcater  pressure  changes,  both  Sp  and  S^  change  ;  that  is,  S^ 
and  Sp  both  depend  on  the  pressure  p  : 

Sa  =  ^  function  of  p  alone  ; 
Sp=&  function  of  p  and  of  t. 


296  SEVERAL  VARIABLES  [IX,  §  151 

If  t  is  assigned  a  fixed  value,  sA  =  Sa,  where  A  is  the  position  pf  P 
at  the  time  «  =  A;.     We  have  -^'^* 

dSp^dSp  ^  dSpdp^ 
dt  ct  dp  dt 
where  dSp/dp  is  the  rate  of  change  in  Sp  with  respect  to  p  which  would 
occur  if  t  alone  could  be  kept  constant,  i.e.  the  rate  of  change  of  Sa  with 
respect  to  p,  or  dS^/dp  ;  and  dSp/dt  is  the  rate  of  change  of  Sp  with 
respect  to  t  which  would  exist  if  p  alone  were  constant.*  The  equation 
therefore  shows  that  the  actual  rate  of  change  of  Sp  with  respect  to  t  is 
equal  to  the  rate  at  which  Sp  would  change  if  p  were  constant  plus  the 
rate  at  which  Sa  changes  with  respect  to  p  times  the  rate  of  change  of  p 
with  respect  to  t.  All  of  these  concepts  can  be  illustrated  by  the  speeds 
of  water  particles  near  the  nozzle  of  an  ordinaiy  garden  hose  as  the  water 
is  turned  on  or  off. 

EXERCISES  LXII.  — APPLICATIONS  OF  TOTAL  DERIVATIVES 

1.  Find  dz  when  z  =  x^  +  4  y^.  Hence  find  dy/dx  for  the  point  x  =  1, 
y  =  2  on  the  curve  x2  + 4  2/2  —  17.  Find  dy/dx  for  that  curve  of  the 
family  a;^  +  4  2/2  =  ^  which  passes  through  x  =  2,  y  =  1,  at  that  point. 

2.  Find  dy/dx  for  that  curve  of  the  family  xy  =  k  which  passes 
through  the  point  x  =  2,  y  =  3,  at  that  point. 

3.  Find  dy/dx  for  that  curve  of  the  family  x^  -\- y^  —  Zxy  =  ]c  which 
passes  through  the  point  (1,  1)  at  that  point. 

4.  For  steam,  it  is  found  by  experiment  that  pv^'^"^^  =  const.,  for 
adiabatic  expansion.  Find  dp/dv  and  dd/dv,  where  d,  v,  p  denote  the 
temperature,  volume,  and  pressure,  respectively,  and  pv  =  kd. 

5.  The   strength  of  a  beam  is  proportional  to  bd?/l,  where  b  is  the 
breadth,  d  the  depth,  and  I  the  length  of  the  beam.    Discuss  the  effect  ^ 
upon  the  strength  of  changes  in  each  dimension  separately  ;  the  effect  of 
simultaneous  changes  in  6  and  d  when  I  is  constant. 

6.  In  the  beam  of  Ex.  5  if  &  and  d  are  changed  while  1  is  constant, 
find  a  relation  connecting  b  and  d  if  the  strength  remains  unchanged. 
Find  the  rate  of  change  of  6  with  respect  to  d  under  these  circumstances. 

*  For  this  reason,  the  partial  derivative  dSp/dt  is  often  called  a  "particle  " 
derivative :  it  is  in  this  case  the  fictitious  rate  at  which  Sp  would  change  if 
the  flow  were  steady,  as  P  moves  along  the  pipe.  The  other  derivative 
dSp/dp  may  be  replaced  by  dSJdp. 


IX,  §  151]  PARTIAL  DERIVATIVES  29? 

7.  The  amount  of  deflection  Z)  of  a  rectangular  beam  under  a  load 
is  proportional  to  l^/bd^,  in  the  notation  of  Ex,  5.  Find  the  rates  of 
change  of  D  with  respect  to  each  dimension  separately.  Find  a  relation 
between  I  and  d  for  which  D  is  constant  while  b  is  constant ;  and  find 
dl/dd  in  this  case.  Find  a  relation  between  I  and  b  for  which  D  and  d 
are  constant  and  find  db/dl  in  this  case. 

8.  For  the  beam  of  Ex.  7  show  that  if  b  is  constant, 

dD  =  (3  l''/bd^)[_dl  -  il/d)dd]. 

9.  The  collapsing  pressure  of  a  boiler  tube  is  given  by  Fairburn  as  pro- 
portional to  t^/ld  where  t,  1,  d,  respectively,  denote  the  thickness  of  the 
material,  the  length,  and  the  diameter,  of  the  tube.  Show  how  the  col- 
lapsing pressure  changes  with  respect  to  changes  in  t  and  d. 

10.  The  resistance  J?,  due  to  water  friction  for  a  boat  in  still  water,  is 
proportional  to  S-D-^^,  where  S  is  the  speed  and  D  is  the  displacement. 
Show  how  the  resistance  changes  when  S  changes  ;  when  D  changes. 

11.  If  the  boat  of  Ex.  10  is  loaded  more  heavily,  D  increases  ;  but  S 
is  usually  decreased.     Find  dS/dD  if  i?  is  kept  constant. 

12.  The  temperature  at  points  of  a  certain  square  plate  OABC  varies 
inversely  as  1  -|-  r-,  where  r  is  the  distance  from  O.  The  temperature  at 
O  is  100°,  Find  the  rate  of  change  of  the  temperature  (a)  along  the 
diagonal  OB,  (ft)  along  AC  at  the  center  of  the  plate;  (c)  along  a  verti- 
cal line  through  the  center  of  the  plate. 

13.  If  M  is  a  function  of  three  variables,  such  as  the  density  or  the 
temperature  at  points  of  a  solid,  the  rates  of  change  of  u  in  the  directions 
of  the  coordinate  axes  are,  respectively,  du/cx,    du/cy,    cu/cz. 

If  s  is  a  variable  distance  along  a  line  making  angles  a,  /3,  y  with  the 
coordinate  axes,  show  that  the  rate  of  variation  of  «  along  this  line  is 

du      du  dx  ,  du  dii  ,  du  dz      du  „^„  ^  ,  du  „„^  „  ,   ^m  „„„ 

—  = 1 2-1 —  =  —  cos  a  -^ cos  p  -\ cos  7. 

ds      ex  ds      dy  ds      cz  ds      dx  dy  dz 

^  14.  In  a  spherical  shell  of  inner  radius  5  and  outer  radius  10,  the  tem- 
perature decreases  uniformly  from  100°  at  the  inner  surface  to  0°  at  the 
outer.  Show  that  the  rate  of  variation  of  the  temperature  along  a  radius, 
at  right  angles  to  a  radius,  along  a  line  inclined  45°  to  a  radius  at  their 
point  of  intersection  are,  respectively,  —  20,  0,  —  IOV'2. 

15.  From  the  value  of  dy/dx  found  in  Example  3,  §  151,  show  that  the 
equation  of  the  tangent  to  a  plane  curve  whose  equation  is  given  in  im- 
plicit form,  f(x,  y)  =  0,  is  (x  -  Xp)  (df/dx)p+  (y  -  yp)  (cf/cy)p  =  0, 
where  (xp,  yp)  is  the  point  of  tangency,  and  where  the  values  of  the 
derivatives  are  to  be  taken  at  that  point. 


298 


SEVERAL  VARIABLES 


[IX,  §  151 


PART   II.     APPLICATIONS   TO   PLANE   GEOMETRY 
152.   Envelopes.     The  straight  line 
(1)  y  —  lex  —  l?, 

where  k  is  a  constant  to  which  various  values  may  be  assigned, 
has  a  different  position  for  each  value  of  k.  All  the  straight 
lines  which  (1)  represents  may  be  tangents  to  some  one  curve. 

If  they  are,  the  point  P^, 
(a-,  y)  at  which  (1)  is  tan- 
gent to  the  curve,  evidently 
depends  on  the  value  of  k : 

(2)  x  =  <l>(k),  y=xl^(k); 

these  equations  may  be  con- 
sidered to  be  the  parameter 
equations  of  the  required 
curve.  The  motive  is  to 
find  the  functions  <f>{k)  and 
ij/(k)  if  possible. 

Since  P^.  lies  on  (1)  and 
on  (2),  we  may  substitute 
from  (2)  in  (1)  to  obtain : 

Fig.  67  (3)     xl;{k)  =  k<i>{k) -k', 

which  must  hold  for  all  values  of  k.     Moreover,  since  (1)  is 
tangent  to  (2)  at  P^,  the  values  of  cly/clx  found  from  (1)  and  x. 
from  (2)  must  coincide : 

from  (1)     dx_\  from  (2)        <h'(k)  ' 

To  find  4>(k)  and  i{/(k)  from  the  two  equations  (3)  and  (4),  it  is 
evident  that  it  is  expedient  to  differentiate  both  sides  of  (3) 
with  respect  to  k : 

(3*)  ^p'Qc)  =  H'(k)  +  <|>{k)-2k^, 


(^)      k  =  '-^1  =  —1 

^   ■'  clx}  from  (1)     dx_\ 


or  kcj>'{k)  =  yl,'(k). 


IX,  §152]  GEOMETRY  — ENVELOPES  299 

this  equation  reduces  by  means  of  (4)  to  the  form 

(5)  0  =  0  +  </>(^•)  -2k,  or  <^{k)  =  2k, 
and  then  (3)  gives 

(6)  4,(Jc)  =  k(2k)-k^  =  k\ 

Hence  the  parameter  equations  (2)  of  the  desired  curve  are 

(7)  x  =  2k,     y  =  k\ 

and  the  equation  in  usual  form  results  by  elimination  of  k : 

(8)  y  =  f. 

It  is  easy  to  show  that  the  tangents  to  (8)  are  precisely  the 
straight  lines  (1) 

The  preceding  method  is  perfectly  general.     Given  any  set  of  curves 

(1)'  F{x,  y,  k)  =  0, 

where  k  may  have  various  values,  a  curve  to  which  they  are  all  tangent 
is  called  their  envelope  ;  its  equations  may  be  written 

(2)'  x  =  <t>{k),y  =  f{k-); 

whence  by  substitution  in  (1)', 

(3)'  Fi<p{k),^|^(k),k■]  =  0, 

for  all  values  of  k.    Differentiating  (3)'  with  respect  to  A, 

/3*-)/  dF{x,  y,  k)  _dF  dx      tF  dy      ^F _q 

dk  ~  dx  dk      dy  dk      dk  ~ 

Moreover,  since  (1)'  is  tangent  to  (2)', 

dx    '    dy       rfvBj  from  (1)'     dicj  from  (2)'       dk   '  dk' 

whence  (3*)'  reduces  to  the  form 

(5)'  ^  =  0; 

ck 

and  then  (3)'  and  (5)'  may  be  solved  as  simultaneous  equations  to  find 
0  (k)  and  ^  (k)  as  in  the  preceding  example. 


300 


SEVERAL  VARIABLES 


[IX,  §  152 


The  envelope  may  be  found  speedily  by  simply  writing  down  the  equa- 
tions (1)'  and  (5)',  and  then  eliminating  k  between  them.  It  is  recom- 
mended very  strongly  that  this  should  not  be  done  until  the  student  is 
familiar  with  the  direct  solution  as  shown  in  the  preceding  example. 

153.  Envelope  of  Normals.  Evolute.  The  normal  to  the 
curve  y  =  X'  at  a  point  x  =  'k  is 

1 


(1) 


y-k-  = 


2  k 


(x-Jc). 


Fig.  U8 
If  these  lines,  for  all  values  of  k,  are  tangent  to  some  one  curve : 
(2)  x  =  <l>(k),    y  =  ^(k), 

direct  substitution  gives 


(3) 


xj/(k)  —  k'^  = 


2  k 


[<^(fc)-A;], 


for  all  values  of  k.     Differentiation  with  respect  to  k  gives  * 
(3*)         ^•{k)-2k  =  ^X_<t>{k)-k-]-^l.i>'{k)-ll 

*  The  reason  for  this  differentiation  is  brought  out  in  the  example  of  §  152. 
Notice  that  the  equations  here  are  numbered  to  correspond  exactly  to  the 
equations  of  §  152. 


IX,  §  153]  GEOMETRY  —  EVOLUTES  301 

Moreover,  since  (1)  is  tangent  to  (2) 

whence  (3*)  reduces  to 

(5)  0-2A:  =  A_[^(fc)_A;]-J^(0-l). 

Solving  for  ^(^•),  we  find : 

(6)  «^ (fc)  =  ^^  +  2  ^-2^  -  2  A;  -  ^^  =  -  4  A;3. 

[t  follows  from  (3)  that 

[7)  ^  (A;)  =  Ar'  -  i^  [<^  {k)-k^  =  3 Tc^^  1/2, 

whence  the  equations  of  the  new  curve  are 

[8)  a;  =  -  4  Ar',     y  =  3  A;-  +  1/2. 

We  might  proceed  to  eliminate  A:,  as  in  the  example  of  §  152, 
in  order  to  express  the  equation  of  the  new  curve  in  usual  form  ; 
but  when  the  elimination  is  at  all  diificult,  as  it  is  here,  it  is 
best  to  keep  the  equation  in  the  parameter  form  (8).  The 
'raph  may  be  plotted  from  these  equations  as  usual.  The 
iccurate  construction  of  a  few  normals  to  the  given  curve  is  a 
jreat  assistance  in  drawing  this  graph. 

The  new  curve  is  called  the  evolute  of  the  given  curve;  the 
*iven  curve  is  called  an  involute  of  the  new  one :  the  evolute  to 
xny  curve  is  the  envelope  of  its  normals.  (See  §  154  and  Ex.  4, 
p.  172.) 

The  method  used  above  is  perfectly  general,  and  may  be  used  in  any 
problem.     Thus  the  normal  to  a  curve  y  =  /  (jc)  at  a  point  x  —  k  is 

[ly  y-f(k)=-^{x-k). 

If  these  normals,  for  all  values  of  k,  are  tangent  to  the  new  curve 
[2)'  x  =  <l>ik),    y  =  Hk), 


302  SEVERAL  VARIABLES  [IX,  §  153 

direct  substitution,  followed  by  differentiation  with  respect  to  k,  gives 
(3)'  HJc)  -f(k)^-y^[<p(k)  -  A:], 

(3*)'      rw  -f'(k)  =  ^!^^  [<p(ik)  -^^-j~  C^'(^)  -  !]• 

Moreover,  since  (1)'  is  tangent  to  (2)' 
(4)'  ~=     ^1  =^1  "       =xl^i(k)~<p'(k): 

^   ^  f'{k)  dajjfrom(l)'      dicjfrom  (2)'  ^  ^    ''      ^^    ^> 

whence  (3*)'  reduces  to 

(5)'  0-f'{k)=+    f"'^^^    \_<j>{k)-k'\ ^[0-11; 

w  J  \  )        [f'{k)Y  f'{ky         -* 

an  equation  which  might  have  been  found  directly  from  (5)'  of  §  162, 
Solving  (5)'  for  (t>{k),  we  find  :  * 


(6)' 


"^^  ^  ^    /"(/fc)    L     ■'    ^   ^      /'(A:)J 


whence,  from  (3)' 

(7)'  ^(k)  =f(k)  -^lci>(k)-k]  =f(k)  +  '^  +  lf'[\^^'. 

Denoting /(fc)  by  y^,  f'(k)  by  m*  (the  slope  at  x=k),  f"(k)  by  bk  (the 
flexion  at  x  =  k),  the  equations  of  the  evolute  may  be  written  in  the 
form : 

CSV                      ,      »Wfc(l  +  mfc2)                               1+mjl 
w  x  =  k V ,  and  2/ =  2/fc  H j ^ 

"k  "k 

These  equations  may  be  used  to  write  down  the  equations  of  the  evo- 
lute directly ;  but  it  is  strongly  recommended  that  the  direct  solution,  as 
above,  be  practiced.  Frequently  the  elimination  of  k  between  the  two 
equations  (8)'  is  rather  difficult,  as  in  the  example  given  above  ;  hence 
the  equations  are  very  often  left  in  the  parameter  form  (8)'. 

*  Notice  that  the  work  breaks  down  at  this  point  iif"{k)  =0.  The  advan- 
tage of  this  direct  solution  is  that  such  special  cases  are  not  so  troublesome  as 
when  the  final  formulas  alone  are  used. 


IX,  §153]  GEOMETRY  — E VOLUTES  303 


EXERCISES  LXm.— ENVELOPES    EVOLUTES 

1.  Show  that  the  envelope  of  the  set  of  straight  lines  y  =  3  kx  —  k^  is 

2.  Find  the  envelopes  of  each  of  the  following  families  of  curves : 
(a)  2/  =  4  kx  —  k*.    Ans.    y^  =  27  x*. 

(6)  y'^  =  kx-  A;2.     Ans.   y=±\x. 


(c)  y  =  kx±  Vl  +  k\     Ans.   x-  +  j/2  =  1. 

(d)  r'  =  ^•-.r-2^•.     Ans.   xy^  =  -l. 

(e)  (x  -  A-)-  +  2/-  =  2  A.     .4«s.    «/2  =  2  x  +  1. 

(/)  4  a;2  +  (y  _  A;)2  =  1  _  fc2.     ^hs.    t/2  +  g  x2  =  2. 

(j^)  X  cos  ^  +  2/  sin  5  =  10.     ^/ts.    x2  +  y-  =  100. 

3.  Show  that  the  normal  to  the  curve  y  =  x^  a,t  any  point  (k,  F)  on  it, 
is  y  —  k^  =  —  (x  —  k)/S  k'-.  Hence  show  that  the  evolute  oi  y  =  x^  is  given 
by  the  parameter  equations  x  =  (^•  —  9  k^)/2,  y  ={lb  k*  +  \)/{Q  k). 

4.  Taking  the  equations  of  an  ellipse  of  semiaxes  a  and  b  in  the  form 
X  —  a  cos  e,  y  =  b  sin  0,  show  that  the  equation  of  the  normal  at  any 
point  is  by  =  ax  tan  6  +  {b-  —  a-)  sin  6.  Hence  show  that  the  evolute  of 
the  ellipse  is  given  by  the  parameter  equations  ax=(a2— 62)  cos*  $, 
by  =  (62  _  a2)  sin3  0. 

5.  Find  the  evolute  of  the  curve  y^  =  x*. 

6.  Find  the  evolute  of  the  curve  y  =  e*. 

7.  Show  that  the  envelope  of  a  family  of  circles  through  the  origin 
with  their  centers  on  the  parabola  2/2  =  2  x  is  y'^(x  +  1)  +  x^  =  0. 

8.  Show  that  the  envelope  of  the  family  of  straight  lines  ax  +  by  =  I 
where  a  +  6  =  a6,  is  the  parabola  x^/^  +  yV'^i  =  1. 

9.  Show  that  the  envelope  of  the  family  of  parabolas  y  =  xUna- 
«»x2  sec2  a  is  2/  =  1/(4  w)  —  tox2. 

[Note.  If  m  =  g/(2  vd^),  the  given  equation  represents  the  path  of  a 
projectile  fired  from  the  origin  with  initial  speed  vo  at  an  angle  of  eleva- 
tion a.] 

10.    Find  the  evolute  of  the  curve  y  —  (e^  +  e-')/2. 


304  SEVERAL  VARIABLES  [IX,  §  154 

154.  Properties  of  Evolutes.  In  general,  taking  the  equa- 
tions (8)'  of  §  153,  the  equation  of  the  evolute  of  a  given 
curve  y=f{x)  may  be  written: 

(1)  x-x^= ^^- ^,    y-yk=    \    S 

where  Xi,=k,  y^=f(k),  mk  =  f{k)  (the  slope  at  x=^'k),  h^=if"{lk) 
(the  flexion  at  x='k).  The  point  (a;^,  y,)  lies  on  the  given 
curve  y  =  f(x);  the  point  (a;,  y)  lies  on  the  evolute;  the 
normal  at  (x^,  y^)  to  the  given  curve  is  tangent  to  the  evolute 
at  (a;,  y).  Hence  the  distance  B,  measured  along  the  normal, 
from  the  given  curve  to  the  point  of  tangency  on  the  evolute, 
is  given  by  the  equation : 

zy  =  (^  -  .,)'  +  (y  -  y,y  =  «\+  <>'  +  ^^ 

'^k  ^k 

it  follows  that  D  is  precisely  the  radius  of  curvature  (see  §  97, 
and  Ex.  4,  p.  172)  : 

^k 
Hence  the  radius  of  curvature  of  a  curve  is  shown  graphically 
when  the  evolute  is  drawn :  in  Fig.  68,  p.  300,  for  example, 
the  radii  of  curvature  at  A,  B,  C,  D,  0  are  the  lengths  AA', 
BB',  CC,  DD\  00',  respectively.  Notice  particularly  that  the 
change  in  the  radius  of  curvature  can  be  followed  by  the  eye 
very  clearly  by  means  of  the  evolute,  as  the  point  on  the  given 
curve  moves.* 

155.  Center  of  Curvature.  The  point  at  which  the  normal 
to  the  given  curve  is  tangent  to  the  evolute  is  at  a  distance  R 
from  the  given  curve  (along  the  normal)  ;  this  point  is  called 

*  This  is  of  importance  in  laying  out  railroad  curves,  etc.,  where  the 
change  in  the  radius  is  of  great  moment ;  in  particular  the  minimum  value 
of  the  radius  is  often  important. 


IX,  §156]  GEOMETRY -CURVATURE  305 

the  center  of  curvature ;  its   coordinates  are  precisely  the  x 
and  y  of  equations  (1),  §  154.     (See  Ex.  2,  p.  171.) 

The  angle  y8  which  the  normal  makes  with  the  a>axis  is 
shown  in  Fig.  68,  p.  300  ;  from  the  right  triangle  CAC  we  have 

Z  ACC  =  13,  CC  =  E,  CA  =  x  -  x,,  AC'=>/-  y„ 

and  therefore 

(1)  X  —  x\  =  R  cos  /3,  i/  —  y^  =  R  sin  ^ ; 
moreover 

(2)  .V^.  =  tan^  =  -1  =  -^V=* 

x—x^  W/,.  dy^./(Lx\      ax 

since  tan  fi  is  the  slope  of  the  normal  (=  —  l/'»^/fc)  of  the  given 
curve,  and  also  the  slope  of  the  tangent  of  the  evolute. 

The  circle  whose  radius  is  E  (the  radius  of  curvature)  and 
whose  center  is  the  point  (x,  y)  at  which  the  normal  is  tan- 
gent to  the  evolute,  is  called  the  circle  of  curvature ;  its  equa- 
tion is  {X  —  cc)^  +  (  Y—  yY  =  E^,  where  {x,  y)  is  the  fixed  point 
on  the  evolute  and  (X,  Y)  is  the  variable  point  on  the  circle 
of  curvature. 

156.  Rate  of  Change  of  R-  The  rate  at  which  R  changes,  which 
was  mentioned  m  §  154,  can  be  obtained  as  follows.     Since 

B^  =  (x-  Xk)^  +(y-  yk)\ 
we  have 

(2)  RdR  =  (x-  Tk)  (dx  -  dXk)  +  (y  -  y*)  (dy  -  *a-). 
or 

(3)  dH  =  Cx-Xt)c^x+(2/-y*)(?y^ 

Vix-x^y^+iy-yk)-^  ' 
since  ^x  -  Xu)  dXk  +  (y  -  yk)  dy^  =  0,  by  (2),  §  165. 

But  since  (y  -  yt)/(x  -  x^)  =  dy/dx,  by  (2),  §  155, 


(4)  dR  = _^^*,     =  ^===  =  Vdx:^  +  dy^. 


and  since  Vdx:^  +  dy^  =  ds,  where  s  is  the  length  of  arc  of  the  evolute, 

X 


306  SEVERAL  VARIABLES  [IX,  §  156 

(5)        dR  =  ds,     or  r    ~   'di?  =  ("'"'"rfs,     or  ^2  -  ^i  =  Sa  -  Si  ; 

that  is :  the  rate  of  growth  of  the  radius  of  curvature  is  equal  to  the  rate 
of  growth  of  the  arc  of,  the  evolute  ;  and  the  difference  between  two  radii 
of  curvature  is  the  same  as  the  length  of  the  arc  of  the  evolute  which 
separates  them. 

This  fact  gives  rise  to  an  interesting  method  of  drawing  the  original 
curve  (the  involute)  from  the  evolute  :  Imagine  a  string  wound  along  the 
convex  portion  of  the  evolute,  fastened  at  some  point  (say  D',  Fig.  68, 
p.  300)  and  then  stretched  taut.  If  a  pencil  is  inserted  at  any  point 
(say  C,  Fig.  68)  in  the  string,  the  pencil  will  traverse  the  involute  as  the 
string,  still  held  taut,  is  unwound  from  the  evolute. 

157.  Illustrative  Examples. 

Example  1.     The  evolute  of  the  curve  y  =  x"^  was  found  in  §  153  to  be 

x  =  -4F,     ?/  =  3A;2+l/2. 

The  radius  of  curvature  of  the  given  curve  at  the  point  (x^  =  k,  yk  =  k^) 
is  therefore 


B  =  V(x  -  ky  +(y-  i-2)2  =  V(-  4  A;3  -  ky  +  (2  ^2  +  1/2)2 
=  H4  A;2  +  1)3/2. 
The  rate  of  change  of  B  with  respect  to  k  is 

—  =  6A;(4A;2  + 1)1/2; 
dk 

and  JB  is  a  maximum  or  a  minimum  only  where  this  rate  is  zero,  i.e. 
where  A;  =  0.  Since  dB/dk  is  negative  when  ^  <  0,  and  positive  when 
^>0,  it  follows  that  i?  is  a  minimum  when  k  =  0,  i.e.  at  the  point  0  in 
Fig.  68,  p.  300.  Tiie  value  of  B  at  this  point  is  Bo  =  1/2 ;  this  is  also 
evident  in  the  figure. 

Example  2.     To  find  the  evolute  and  radius  of  curvature  of  the  cycloid ; 

,,v  (x  =  a(t  — sin  t), 

^  [y  =  ail -cost). 

The  slope  m^  at  a  point  (x*.,  y^)  where  «  =  A;  is 

dyk 
^   ^  dyk^  dt  ^       asint       ^     smt     ^^^^i. 
*      dx^      dxk     a(l  — cos<)      1  — cos«  2 

dt 


IX,  §  157]  GEOMETRY  — CURVATURE 

and  the  second  derivative  b^  =  cPy,,/dx^k  is 


307 


dm^      cos  <  ( 1  —  cos  t)  —  sin''  t 


_  dnik  _  dt 
*  ~  dx^  ~  dx^ 


(1  -cosO^ 


dt 


a{\-co&t)  a{l-cos,tY     Aasm\t/2) 

It  follows  that  the  evolute  is  given  by  the  equations  : 
X 

1  +  mr . 


.Tfc  -  "'*C^  +  ""■■'^)  =  rt(<  -  sin  t)+2a  sin  t  =  a{t  +  sin  t). 


y  =  Vk+- 


hk 


a{\  —  cos «)  —  2  rt(l  —  cos  t)  =  —  a{\  —  cos  t) 


which  is  another  cycloid  of  the  same  shape  and  size,  with  its  vertices  at 
the  points  y  =—2a,  x  —  wa,  3  ira,  etc.,  as  shown  in  Fig.  69. 


y 

<\ 

Inv 

A 

^ —     *^ 

A' 

/ 

^^' 

"""'-■-^    X 

Fig.  69 


The  radius  of  curvature  is  given  by  the  equation 

ii;2  =  (x  -  a-4)2  +  (2/ -  yj2  ^  (2flsin02  +  (- 2  a(l  -  cosi))^ 
-  8  a2(l  -  cos  t). 


whence 


i?=4a^Lzi|2ii  =  4a^/8in2(|)=4«sinQ) 


The  value  of  i?  at  0  is  given  by  «  =  0  :  J?o  =  0 ;  the  value  of  II  at 
C  is  given  by  «  =  tt  :  7?c  =  ^  «•  Since  the  arc  OC^  of  tlie  evolute  is 
equal  to  the  difference  of  these  values  of  7?,  we  have  arc  OC  =  4  a;  hence 
the  length  of  a  whole  arch  of  the  cycloid  is  2  00'  =  8a  (Ex.  16, 
p.  155). 


308  SEVERAL  VARIABLES  [IX,  §  157 

EXERCISES   LXI v.  — PROPERTIES   OF  E VOLUTES 

1.  Find  the  general  equation  of  the  circle  of  curvature  for  the  curve 
y  =  x3.     Draw  it  for  the  points  (1,  1),  (2,  8),  (1/2,  1/8). 

2.  Find  the  radius  of  curvature,  the  circle  of  curvature,  and  the 
evolute,  for  any  point  on  the  curve  a;  =  4  cos  6,  y  =  sin  d. 

3.  Find  the  radius  of  curvature,  the  circle  of  curvature,  and  the 
evolute  (in  parameter  form)  for  each  of  the  following  curves  at  any  point : 

(a)    (^  =  si^^.  (fZ)    (^  =  «' 

[y  =  2  cos  d.  \y  =  cos  t. 

,,.     fx  =  sec  e,  .  .    jx  =  cost  +  t  sin  t, 

\y  =  tan  0.  [y  =  sin  t  —  t  cos  t. 

4.  Find  the  minimum  value  of  the  radius  of  curvature  for  the  curve 
y  =  x^     Ans.  E  =  (3/5)  </4/6. 

5.  Show  that  the  length  of  one  quarter  of  the  evolute  of  an  ellipse  is 
(a3  _  b^)/ab. 

6.  Show  that  the  curvature  of  a  curve  at  any  point  of  inflexion  is 
zero, 

7.  The  curvature  /i  =  1/R  is  obtained  by  multiplying  the  flexion  b 
Hby  the  corrective  factor  (l  +  ?)i2)-3/2_  show  that  this  corrective  factor  is 
equal  to  cos^  «,  where  a  is  the  angle  between  the  given  curve  and  the  » 
axis.     (See  §  97,  p.  169.) 

8.  Show  directly  from  the  definition  of  §  97  that  K=da/ds=b  cos^  a. 
[Hint,    m  =  tan   a,  hence  dm  =  sec^  a  da  ;  but  dx/ds  =  cos  a  and 

dm/dx  =  5.] 

9.  Find  the  equation  of  the  evolute,  and  its  length  of  arc,  for  the 
tractrix,  y  =  a  sin  6,  x  =  a  log  cot  (0/2)  —  a  cos  d. 

10.  The  evolute  of  the  equiangular  spiral  p  =  «e*^  is  an  equal  curve 
turned  through  an  angle  ir/2  +  k  log  k.     Determine  the  length  of  arc. 

11.  When  a  thread  is  suspended  from  one  cusp  of  a  horizontal  inverted 
cycloid,  the  thread  carrying  a  weight  at  the  free  end  and  having  a  length* 
equal  to  twice  the  height  of  arch  of  the  curve,  show  that  if  the  weight  be 
made  to  oscillate  so  that  the  thread  winds  up  on  the  curve,  it  will  describe 
an  equal  cycloid. 


IX,  §  158]  SINGULAR  POINTS  309 

12.  Construct  the  evolutes  of  the  curves, 

(a)  y  =  sin  x,     (6)  y  =  tan  x,     (c)  y  =  a  cosh  (x/a). 

13.  The  lemniscate  (^x^  +  y-)^  =  a'^(x2  —  y'^)  may  be  written  : 


X  =  a  cos  «  Vcos  2t;  y  =  a  sin  t  Vcos  2  «, 
Show  that  the  e volute  is  (x2/3+j/2/3)2  (■a;2/3  _  j,2/3)  =  4  02/9. 

158.  Singular  Points.  The  tangent  to  a  curve  whose  equa- 
tion is  given  in  explicit  form  y  =  f(x),  where /(a;)  is  single- 
valued,  can  neither  be  vertical  nor  fail  to  exist  if  dy/dx—f'(x) 
exists.  If  the  equation  of  the  curve  is  given  in  the  implicit 
form, 

(1)  /(^,2/)  =  0, 

the  slope  of  the  tangent,  m=dy/dx,  is  given  by  the  equation 
(Example  3,  §  151,  p.  294) : 

(2)  ^  +  ^^  =  0, 
dx      dy  dx 

which  can  be  solved  for  dy/dx  as  in  §  151,  unless  df/dy  =  0. 
At  a  point  S  at  which  df/dy  =  0,  if  df'dx^  0,  the  tangent  is 
vertical ;    if  both  df/dy  and  df/dx  are  zero  at  S,  the  equation 

(2)  is  practically  useless,  and  the  tangent  may  be  indeterminate. 
For  this  reason,  a  point  S  on  the  curve  (1)  for  which  df/dx 

and  df/dy  both  vanish  is  called  a  singular  point ;  such  points 
may  be  found  by  solving  the  equations 

(3)  ^  =  0,       ^^  =  0, 
^  ^  dx       '       dy 

as  simultaneous  equations  for  x  and  ?/.  The  points  thus  found 
may  not  lie  on  the  curve  (1) ;  if  not,  they  should  be  discarded. 

Notice,  however,  that  any  pair  of  solutions  (a,  ?>)  of  (3)  are  the 
coordinates  of  a  singular  point  of  the  contour  line  of  the  surface  z=f(x,  y) 
cut  out  by  the  plane  z  =  c,  where  /(a,  b)  =  c. 

Usually  a  due  amount  of  care  in  plotting  the  curve  near  tlie 
singular  point  will  indicate  its  nature.    A  detailed  discussion  is 


310  SEVERAL  VARIABLES  [IX,  §  158 

given  in  advanced  texts  on  Calculus.*  The  points  of  (3)  for 
which  df/dy  alone  vanishes  should  also  be  inspected  carefully 

159.   Illustrative  Examples. 

Example  1.     Examine  the  curve  x^  +  y^—Sxy  =  Ofor  singular  points. 

In  this  example /(x,  y)  =  x^  +  y^  —  S  xy  =  0.  In  §  27,  p.  45  we  found 
dy/dx  for  this  curve  by  a  process  which  amounts  to  the  same  thing  as 
writing 

dx     dydx  dx 

This  equation  can  be  solved  for  dy/dx  [or  for  dx/dyl  imless 

x^  —  y  =  0,  y^  —  x  =  0. 
These  equations  have  the  two  pairs  of  solutions  (a;  =  0,  y  =  0)  and 
(x  =  1,  2/  =  1).  The  point  (0,  0)  lies  on  the  curve,  and  is  therefore  a 
singular  point.  A  careful  figure,  drawn  as  in  Ex.  12,  p.  63,  shows  that 
the  curve  crosses  itself  at  this  point,  and  has  no  single  tangent  ;  such  a 
point  is  called  a  double  point.     (See  Tables,  III,  I5.) 

The  point  (1,  1)  does  not  lie  on  the  given  curve  since  /(I,  1)  =1  +  1  —  3 
=  —  1.  But  it  does  lie  on  the  contour  line  of  the  surface  z  =  x^  +  y^—Sxy 
cut  out  by  the  plane  z  =  —  1.  A  careful  graph  of  this  contour  line 
x^  +  y^  —  Sxy  =  —  I  reveals  the  fact  that  there  is  no  other  point  on  the 
curve  near  (1,  1),  although  there  is  another  portion  of  the  curve  some 
distance  away  ;  such  a  point  is  called  an  isolated  point.  The  plotting 
of  the  figures  is  facilitated  by  first  rotating  the  x?/-axes  through  45°. 

Example  2.     Examine  the  curve  y^  =  x^  for  singular  points. 

Here/(a;,  y)  =  y^  —  x-  =  0,  and  we  write  : 

^£  +  ^f.^y=^2x  +  3y^^  =  o, 

dx      dy  dx  dx 

an  equation  which  determines  dy/dx  [or  dx/dy"]  except  when  x  =  y  =  0. 
This  point  (0,  0)  lies  on  the  given  curve  ;  hence  it  is  a  singular  point. 
Careful  plotting  (see  Tables,  III,  A)  near  the  point  indicates  that  the 
curve  has  a  sharp  corner  at  this  point.  At  any  other  point  dy/dx 
=  2x/(Sy-)  =2/(3x1/3).  As  x  approaches  zero  from  either  side,  this 
quantity  becomes  infinite.  Hence  the  tangent  approaches  a  vertical  posi- 
tion as  X  approaches  zero  from  either  side.  A  corner  is  called  a  cusp  if 
the  two  branches  of  the  curve  which  meet  there  have,  as  here,  a  conmion 
tangent  line. 

*  See,  e.g.,  Goursat-Hedrick,  Mathematical  Analysis,  Vol.  I,  p.  110. 


IX,  §  160]  ASYMPTOTES  311 

Example  3.     Examine  the  cycloid 

X  =  a{t  —  sin  <),     y  =  a{\  —  cos  i) 
for  singular  points. 

The  value  of  dy/dx  [or  dx/dy"]  is  given,  as  above,  by  the  equation 

dy  _  dy/dt  _      a  sin  <      _     sin  t 
dx      dx/dt     a(l— cos  0      1  —  cos  «' 

unless  sin  «  =  0  and  1  —  cos t  —  0;  these  equations  are  both  satisfied 
when  «  =  0,  ±  2  tt,  etc.,  (not  at  t  —  ir).  Hence  the  points  where  t  —  0, 
[i.e.  (x  =  0,  2/ =  0)],  «  =  27r  \_i.e.  (x  =  2  7ra,  ?/=0)],  etc.,  are  singular 
points.  It  results  from  the  rules  for  indeterminate  forms  (§  136,  p.  263) 
that 

lim  ^  _  lim  1  -  cos  ^  ^  lim  sin  <  _  ^  . 


t^ 


dy 


hence  dx/dy  approaches  zero  as  t  approaches  zero  [i.e.  as  (x,  y)  approaches 
(0,  0)]  ;  therefore  the  tangent  becomes  more  and  more  nearly  vertical  as 
we  approach  the  singular  point  (0,  0)  from  either  side  ;  the  singular  points 
of  a  cycloid  are  therefore  cusps. 

When  the  equations  of  a  curve  are  given  in  parameter  form,  the  singu- 
lar points  can  be  located  as  in  this  example,  by  finding  the  common  solu- 
tions of  the  equations  dx/dt  =  0,  dy/dt  =  0  ;  and  it  is  usually  possible  to 
determine,  by  the  rules  for  indeterminate  forms,  what  happens  to  the 
tangent  as  that  point  is  approached. 

160.  Asymptotes.  The  search  for  the  asymptotes  of  a  curve 
is  often  facilitated  by  our  knowledge  of  the  Calculus. 

Vertical  or  horizontal  asymptotes  are  usually  best  found  by 
the  purely  algebraic  methods  of  analytic  geometry.  Thus  if 
f(x)  is  a  fraction,  the  curve  y  =f{x)  has  a  vertical  asymptote 
ic  =  fc  if  a  factor  of  the  denominator  vanishes  w^hen  x='k.  (See, 
however,  §  139,  p.  268).  If  f{x)  has  a  factor  tan  x  or  log  x  or 
sec  X,  •■■,  y  =  f(x)  may  have  a  vertical  asymptote  at  any  point 
where  that  factor  becomes  infinite.  Useful  rules  for  horizontal 
asymptotes  result  by  interchange  of  x  and  y. 

If  the  asymptote  is  neither  horizontal  nor  vertical,  these 
elementary  means  are  insufficient.     If  the  tangent  to  a  given 
curve : 
(1)  y-yp  =  '>np{x-xj), 


312 


SEVERAL  VARIABLES 


[IX,  §  160 


at  the  point  P,  (xp,  yp),  approaches  a  fixed  limiting  position 

(2)  y  =  ax  +  b 

as  the  distance  OP  from  the  origin  to  P  becomes  infinite,  the 
line  (2)  is  called  an  asymptote.     This  will  be  true  if  and  only  if 

lim  7np  =  a,  and    lim   (yp  —  mpXp)  =  b, 


where  a  and  b  are  constants.  The  value  of  m^  can  be  com- 
puted by  any  of  our  usual  methods  and  then  lim  m^  can  be 
found  if  it  exists.  In  this  work  it  is  useful  to  notice  that  the 
ratio  [y/x']p  also  ajyproaches  a  if  there  is  actually  an  asymptote 
(2)  which  is  not  vertical. 

Example  1.     Examine  curve  x^  +  y^  —  3  xy  =  0  for  asymptotes.     The 
method  used  in  Ex.  1,  p.  310,  gives 


hence 


mp  =  m    =^"1   ; 
dx]p     x  —  y^Ap 


lim  trip  =  lim 


p 


=  lim 


JL 


since  1/xp  approaches  zero,  and  {_y/x']p  approaches  a  if  a  exists.     Since 
lim  mp  =  a,  we  have  a  ——  1/a^,  if  a  exists,  whence  a  =  —  1. 

The  equation  of  the  given  curve  may  be  written  in  the  form 


;ir= 


1  +  3 


'?A1 


whence  it  is  evident  that  y/x  does  approach  —  1  as  x  becomes  infinite. 
Finally  the  expression  yp  —  mpXp  becomes 


2xy  —  x^  —  y'' 


-y^l    _   -xy-|    _  y/x        1    . 

^       Jp     x-y'Up      (y/xy^-l/xJp' 


hence 


lim   (yp  —  nipXp)  =  lim 


IX,  §  161]  CURVE  TRACING  313 

since  lim  [y/x']p  =—1  and  lim  [1/a;]/.  =  0.  The  values  of  a  and  b  are 
therefore  a  =—  I,  b  =  —  I,  and  the  line  ?/  =—  a;  —  1  is  an  asymptote. 

The  knowledge  of  this  fact  assists  materially  in  drawing  an  accurate 
figure. 

In  general,  an  equation  of  the  form  f(x,  y)  =  0  gives 
m  =  -  (cf/cx)  -  (cf/dy). 

If  f(x,  y)  is  algebraic,  the  value  of  m  can  be  arranged  as  above  in 
powers  of  (y/x)  and  (1/x)  [or  of  (x/y)  and  (l/y)]  ;  and  the  equation 
f{x,  y)  =  0  can  also  be  written  in  terms  of  (y/x)  and  (l/x).  The 
work  in  any  case  is  similar  to  that  of  the  preceding  example. 

161.  Curve  Tracing.  In  order  to  draw  a  curve  whose  equa- 
tion is  given,  it  is  often  desirable  to  find  whether  there  are 
any  asymptotes  or  any  singular  points  before  an  attempt  is 
made  to  draw  the  curve.  It  is  also  useful  to  know  the  posi- 
tions of  any  maxima  and  minima  (§§  37,  47,  13o)  and  of  any 
points  of  inflexion  (§  46,  p.  75).  The  actual  construction  of 
a  few  tangents  is  often  useful,  particularly  at  points  of 
inflexion. 

Elementary  methods  should  not  be  abandoned  ruthlessly. 
Building  up  a  graph  by  adding,  multiplying,  or  dividing  the 
ordinates  of  two  simpler  curves ;  moving  a  curve  vertically  or 
horizontally;  increase  or  decrease  of  scale  on  one  axis  at  a 
time ;  plotting  from  equations  in  parameter  form  ;  in  some 
rare  instances,  rotation  of  axes ;  in  all  cases,  inspection  of  the 
given  equation  for  possible  simplijications :  these  elementary 
methods  are  even  more  fundamental  and  vital  than  the  newer 
ideas  explained  above. 

EXERCISES   LXV.  —  SINGULAR  POINTS,   ASYMPTOTES,   CURVE 
TRACING 

1.  Find  the  asymptotes  A,  and  the  singular  points  .S'  for  each  of  the 
following  curves ;  then  trace  each  curve.  Use  elementary  methods 
whenever  possible,  and  use  the  points  of  inflexion  and  the  extremes,  if 
any  exist.  In  every  case,  try  to  build  up  the  curve  from  simpler  ones ; 
in  most  of  these  exercises,  this  can  be  done. 


314  SEVERAL  VARIABLES  [IX,  §  161 

(a)  y  = ;  A:x  =  a;  no  S. 

a  —  X 

(b)  y^  =  ;  A:  z  =  a;  no  S. 

X—  a 

(c)  2/2  =  x^  —  a;* ;  no  A  ;  S  :  (0,  0) ,  double  point. 

(d)  yi  =  2x^-x^;  A:y=-x+2/3;  S:(0,  0),  cusp. 

(e)  X  =  y(x  -  a)'^ ;  A  :  x  =  a,  y  =  0  ;  no  S. 

(/)  y\2  a-x)  =  x3;  A:x  =  2a;  s :  (0,  0),  cusp. 

(g)    x^y  =  4  a2(2  a-y);  A:y  =  0;  no  /S". 

(h)    y^  =  9  x^  +  x^  ;  A  :  y  =  x  +  3  ;  S  :  {0,  0),  cusp. 

(i)     ?/2(x2  +  1)  =  x2(a;2  -  1)  ;  A:y  =  ±z;  S:(0,0),  isolated. 

U)    2/^(-«  -  2)  ==  x3  -  1  ;  .4  :  a;  =  2,  y  =  ±  (x  4-  1)  ;  no  ,5. 

(*)    y  =  e^  ;  ^  :  ?/  =  0  ;  no  *9. 

(I)    y  =  (e^  +  e-0/2  ;  no  ^  ;  no  S. 

(»n)  y  =  e-"  ;  J. :  y  =  0  ;  no  /S. 

(n)    ?/  =  sec  X  ;  ^  :  y  =  n7r/2,  n  any  odd  integer ;  no  8. 

2.  Show  that  the  curve  y  =  2/(e^  +  e^^)  =  sech  x  is  asymptotic  to  the 
X-axis,  by  building  up  its  graph  from  that  of  1  (l). 

3.  Show  that  each  of  the  curves 

y  =  xe-^,    y  =  x^e-^,   y  =  xV"*,  •••,  y  =  x"e-*, 
is  asymptotic  to  the  x-axis  (See  Exs.  3,  5,  p.  271). 

4.  Show  that  the  curve  1  (a)  has  no  area,  in  the  sense  of  §  111,  be- 
tween x  =  a  and  x  =a  +1,  nor  from  x  =  a  +  ltox  =  Go. 

5.  Show  that  the  curve  1  (6)  has  an  area  between  x  =  a  and  x  =  a  +  1, 
but  not  from  x  =  a  +  ltox  =  oo. 

6.  Show  that  the  curve  of  Ex.  1  (e)  has  no  area  between  x  =  a  and 
X  =  a  +  1 ,  and  has  no  area  from  x  =  a-|-ltox  =  co. 

7.  Show  that  the  curve  y  =  xlogx  ends  abruptly  at  the  origin,  by 
building  up  its  graph.     [See  §  140,  p.  269.] 

8.  Show  that  the  curve  y  =  e~^  sin  x  is  asymptotic  to  the  x-axis. 

9.  Show  that  y  ■=  sin  (1/x)  has  an  infinite  number  of  maxima  and 
minima  near  the  origin  ;  and  that  it  is  asymptotic  to  the  x-axis. 

10.  Build  up  the  graph  oiy  =x  sin(l/x)  from  Ex.  9. 

11.  Show  that  the  curves  y  =  (e*  -f-  e-*)/2=cosh  x  and  y  =  (e*  — e-'^)/2 
=  sinh  X  are  asymptotic  to  each  other,  and  to  the  curve  y  =  e''/2  as  x 
becomes  infinite.     See  Tables,  III,  E. 

12.  Build  up  the  graph  of  y  =  e-^/*' ;  show  that  it  is  asymptotic  to  the 
line  y  =  1. 


[IX,  §  162]  GEOMETRY  OF  SPACE  315 

PART  ni.   gp:ometry  of  space    extremes 

162.   Resume  of  Formulas  of  Solid  Analytic  Geometry. 
(rt)  Distance  between  two  Points  (.t-j,  ^/i,  Zi)  and  (xo,  y.^,  z^ : 

(1)  V(a^2  -  x,y  +  (2/2-  y,y  +  (z,  -  z^\ 

(Jb)  Distance  from  Origin  to  {x,  y,  z)  : 

r  =  V.«-  -f-  y-  -f  z^. 

(c)  Direction  Cosines.  If  «,  /3,  y  denote  the  angles  that  a 
given  line  makes  with  the  positive  directions  of  the  x,  y,  z  axes 
respectively,  then  cos  a,  cos  ji,  cos  y  are  the  direction  cosines 
of  the  given  lines ;  and  we  always  have 

(2)  cos-  a  +  cos-  p  +  cos2  7  =  1. 

If  the  direction  cosines  are  proportional  to  three  numbers  a, 
b,  c,  their  actual  values  are 

cosa  =  —  -,   cos/?  = 


/gx  Va^  +  ft^  +  c^  Va2  +  62^c2' 

cos  y  = 


Va^  +b-  +  c^ 
If  we  indicate  the  direction  cosines  by  single  letters,  say 

(4)  I  =  cos  «,   m  =  cos  13,   n  =  cos  y, 
we  speak  of  the  direction  (Z,  m,  n), 

(rf)  Angle  between  Two  Directions.     The   angle  6  between 
the  directions  (I,  m,  n)  and   (V,  m',  n')   is  given  by 

(5)  cos  d  =  ll'  +  mm'  +  nn'. 
The  directions  are  parallel,  if 

(6)  W  +  ??m'  +  ?in'  =  1. 
They  are  perpendicular,  if 

(7)  IV  +  mm'  +  «n'  =  0. 


316 


SEVERAL  VARIABLES 


[IX,  §  162 


(e)  The  Plane.  If  p  is  the  length  of  the  perpendicular  from 
the  origin  upon  a  plane  and  (I,  m,  n)  is  its  direction,  the  plane 
is  denoted  by  {I,  m,  n ;  p),  and  its  equation  is 

(8)  Ix  +  my  +  nz  =p,  or  x  cos  a  +  y  cos  /3  +  z  cos  y  =p. 

Since  the  distance  d  from  the  plane  (I,  m,  n;  p)  to  the  point 
{^x,  yi,  Zi)  is 

(9)  d  =  Ix^  +  myi  +  nzi  —  p ; 

the  form  (8)  is  called  the  distarice  form,  or  the  normal  form, 
of  the  equation  of  a  plane. 

If  the  axial  intercepts  of  a  plane  are  a,  b,  c,  its  equation  is 

(10)  ^  ■  ^  ■  ' 


^  +  f  +  -=l. 
a     0      c 


(11) 


=  0. 


The  plane  through  the  points  (x^,  y^,  %),  (^2,  2/2;  ^2),  (^3,  Vz,  ^3) 
^^  a;,  ?/,  2,  1 

^•1,  Vi,  ^1,  1 

^2j  y2i  ^2)  1 
^3J  ^SJ  ^3>   1 

The  general  equation  of  the  plane  is  the  general  equation  of 
the  first  degree,  namely  : 

(12)  Ax  +  By-i-Cz  +  D  =  0. 

If  the  direction  of  the  normal  to  the  plane  from  the  origin 
is  {I,  m,  n),  and  its  distance  from  the  origin  is  p,  we  have 

A  B 


I  =■ 


(13) 


^A'  +  B'+C 
C 


p  = 


^A'  +  B'+C' 
-D 


-JW+W+C'  VA'  +  B'+C' 

The  angle  between  two  planes  is  the  angle  between  their 
*  The  definition  of  a  determinant  is  given  in  the  Tables,  II,  C,  5. 


IX,  §  162]  GEOMETRY  OF  SPACE 

normals ;  it  is  given  by 
(14)         cos 


317 


AA'  +  BB'  +  CO' 


V(^2  +  B^  +  C'){A'^  +  B''  +  C"2) 

The  planes  are  ijarallel,  if 

A/A'  =  B/B'-^C/C') 

they  are  perpendicular,  if 

AA'  +  BB'+CC'  =  0. 

The  distance  from  the  plane   ^.r  +  jB?/  4-  Cfe  +  Z)  =  0  to  the 
point  (xi,  ?/i,  Zi)  is 

^A^^W+U' 


(15) 


(/)  The  Straight  Line.     In  general,  a  straight  line  is  repre- 
sented by  the  intersection  of  two  planes  : 

(16)         Ax-\-By+Cz  +  D  =  0,     A'x  +  B'l/  +  C'z  +  D'  =  0. 

The  direction  cosines  of  the  line  are  given  by  the  proportion 


(17) 


l:m:n  = 


BC 
B'C 

CA 

•    CA' 

AB 
A'B' 

together  with  the  principle  (3). 

The  equations  of  the  straight  line  joining  the  points  (xi,  y^,  z^) 
and  (o:.2,  y-2,  z^)  are 


(18) 


X  —  Xi  _y  —  Vi  _  z  —Zi 

X.,  -Xi        ?/2  -  ?/i        Z,  -  2i 


The   line   through   the    point   (x^,  y^,  Zj)   in    the    direction 
(J,  m,  n),  is 


(19) 


x  —  Xt_  _  y  —  ?/i  ^  z  —  Zi 
I  m  n 


318  SEVERAL  VARIABLES  [IX,  §  163 

(gr)  Cluadric  Surfaces.     Equations  of  the  Second  Degree. 
Spheres,  center  (a,  h,  c),  radius  r : 
(20)  (^x-ay  +  (y-by+(z-cy^r^. 

Cones,  vertices  at  origin : 

(Imaginary,  if  all  signs  are  alike ;  otherwise  real,  and  sections 
parallel  to  one  of  the  reference  planes  elliptic.) 

Ellipsoids  and  hyperboloids,  centers  at  origin  (Tables,  III,  N)  : 

(22)  ±^^±f,±'~  =  l. 

a~      ¥     r 

All  signs  on  the  left  +,  ellii^soid. 

One  sign  on  the  left  — ,  hyperholoid  of  one  sheet 

Two  signs  on  the  left  — ,  hyperholoid  of  two  sheets. 

Three  signs  on  the  left  — ,  imaginary. 

Paraboloids,  vertices  at  the  origin  (Tables,  III,  ^4,5)  : 

(23)  ±S4  =  '^- 

Like  signs,  elliptic  paraboloid;  unlike,  hyperbolic  paraboloid. 

163.   Loci  of  One  or  More  Equations  in  Three  Variables. 

A  single  equation  in  three  variables, 

(1)  J^Ccc,  2/,  s)=0, 

represents,  in  general,  a  curved  surface  in  space.  If  z  is  given 
a  series  of  constant  values  a^,  a^,  a^,  •••  successively,  the  coor- 
dinates X,  y  will  satisfy  the  equations  of  the  curves 

(2)  F(x,y,a^)=0,     F(x,y,a,)=0,     F(z,  y,  a,)  =  0,  ••' 
in  which  the  planes 

(3)  z  =  a•^,     z  =  a2,     z  =  a3,     .•• 


IX,  §  163]  GEOMETRY  OF  SPACE  319 

cut  the  surface.  These  curves  are,  in  fact,  contour  lines  on  the 
surface,  aud  the  totality  of  them,  for  all  possible  values  of  z, 
makes  vip  the  surface. 

Two  independent  simultaneous  equations: 

(4)  f{x,y,z)=(i,    ^(U7,  t/,  2)  =  0, 

are  satisfied,  in  general,  by  the  intersection  of  two  surfaces,  and 
therefore  represent  a  curve  in  space. 

TJiree  independent  simultaneous  equations, 

(5)  f{x,y,z)=0,     <l>(x,y,z)=-.0,     ^pifc,  y,  z)  =0, 

are  true,  in  general,  only  at  certain  isolated  points;  those, 
namely,  in  which  the  curve  represented  by  two  of  the  equations 
cuts  the  surface  represented  by  the  third. 

A  single  equation  from  tohich  one  of  the  coordinates  is  missing 
is  a  cylinder  ivith  axis  parallel  to  the  axis  of  the  missing  coordi- 
nate.    Thus 

(6)  /(a?,  2/)=0, 

interpreted  in  space,  is  a  cylinder  parallel  to  the  z-axis.  Its 
trace  on  the  a;i/-plane  is  the  plane  curve, 

(7)  f{x,y)  =  Q,z  =  Q. 

EXERCISES  LXVL— RESUME  OF  SOLID  GEOMETRY 

1.  Find  a  straight  line  tlirough  each  of  the  following  pairs  of  points  ; 
find  its  direction  cosines. 

(a)  (0,  1,  0)  and  (2,  3,  5).  (c)    (4,  1,  -  5)  and  (2,  1,  -3). 

(b)  ( -  1,  2,  -  3)  and  (2,  -  1,  0).        {d)   (5,  3,  7)  and  (5,  -  2,  7). 

2.  Find  the  direction  cosines  of  each  of  the  following  planes : 

(a)  2  X  -  3  y  +  4  0  =  5.  {c)  y-Zz^2. 

iP)  X  -\-y  -\-z  =  0.  {d)  z  =  2x  -y  +  i. 

3.  Find  the  equations  of  a  line  formed  by  the  intersection  of  the 
planes  2  (a)  and  2  (6),  in  the  form  (19),  and  find  its  direction  cosines. 

4.  Proceed  as  in  Ex.  2  for  each  of  the  combinations  formed  by  two  of 
the  planes  mentioned  in  Ex.  2. 


320  SEVERAL  VARIABLES  [IX,  §  163 

5.  Reduce  each  of  the  equations  in  Ex.  2  to  normal  form ;  find  the 
distance  from  each  of  these  planes  to  the  origin. 

6.  Find  the  equation  of  a  plane  through  the  origin  which 
(a)  also  passes  through  the  two  points  of  Ex,  1  (a); 

or  (b)  is  parallel  to  the  plane  2  (a); 

or  (c)   is  perpendicular  to  each  of  the  planes  2  (a)  and  2  (c). 

7.  Find  the  angle  between  each  pair  of  planes  in  Ex.  2. 

8.  Find  the  angle  between  the  direction  specified  by  Ex.  1  (a)  and 
that  specified  by  Ex.  1  (b)  ;  between  the  directions  specified  by  each  pair 
of  lines  mentioned  in  Ex.  1. 

9.  Find  the  center  and  the  radius  of  each  of  the  following  spheres : 
(a)  x2  +  2/2  +  2:2  +  2  X  -  4  2/  +  6  5!  =  2. 

(6)  x^  +  y'^  +  z^+12x  —  y  —  4z  +  40  =  0. 

10.  Find  the  equation  of  a  sphere 

(a)  whose  center  is  (2,  —  1,  4)  and  whose  radius  is  3  ; 

(b)  one  of  whose  diameters  joins  (2,  4,  —  1)  and  (3,  1,  6); 

(c)  whose  center  is  (1,  0,  5)  and  which  passes  through  (3,  1,  —  2). 

11.  Reduce  to  standard  form  and  identify  each  of  the  following  sur- 
faces : 

(a)  a;2 +  41/2  +  ^2  _6x  +  2^  =  6.       (d)  9x^-y^  +  iz^-\-6x  +  10y--25. 

(b)  x2  -  4  j/2  _  6  a;  +  2  s  =  6.  (e)   4x^  -  y- —  4:X  +  6y  =  15. 

(c)  9x2-2/2+4s24.6x+10y  =  10.         {/)  z^  +  9  x"^  -  2  z  +  iy  ^  0. 

12.  Represent  each  of  the  following  equations  or  groups  of  equations 
geometrically  in  space  of  3  dimensions  ;  find  the  trace,  if  any  exists,  on 
each  coordinate  plane,  and  on  each  of  a  series  of  parallel  planes  : 

(a)  z  =  xy.      (b)x^  =  y^  +  z^.        (c)  x^  +  y^  +  z"^  =  1.    (d)y  =  smx. 

(e)  xyz  =  \.     (/)  x2  +  2/2  =  sin z.    (g)  x  +  y  =  e'.  (h)  z  =  e'+y. 

(i)  x^  +  y'^  +  z^  =  i,x  +  y  =  0.       (j)  z-  -  x^  +  2/2,  ^  =  1  -  x. 

{k)  X  =  cos  z,  y=  sin  x.  {I)    x  +  y  =:  e',  y  =  2  x. 

(m)  y  =  z^,  X  =  2/2.  (n)  x-  —  z  —  y,  x  +  y  =  0,  z  +  y=0. 

(o)  x2  —  y2  —4  2,  x  —  2/  =  4,  X  +  2/  =  7. 

(p)  X  +  y  =  z,  y  +  z  =  x,  z  +  x-y. 


IX,  §  164]        TANGENT  PLANE  —  EXTREMES 


321 


164.  Tangent  Plane  to  a  Surface.  Let  Pq  be  the  point 
(xo,  rjo,  Zo)  on  the  surface  z  =f(x,  y).  Let  P„7\  be  the  tangent 
line  at  Pq  to  the  curve  cut 
from  the  surface  by  the 
plane  y  =  yo  and  PqT^  the 
tangent  line  to  the  curve 
cut  from  the  surface  by 
the  plane  x  =  X(t.  The 
plane  containing  these 
two  lines  is  the  tangent 
plane  to  the  surface  at  Pq. 

Since  this  plane  goes 
through  Pi),  its  equation 
can  be  thrown  into  the 
form 

(1)  z-Zf^  =  A(x  -  .To)  -\-B(y-  ?/n). 

If  we  set  y  =  yo  we  find  the  equation  of  PqT^  in  the  form  : 


(2) 

But,  from 
the  form : 

(3) 
Hence 


Z  —  Zn  =  A(X  —  Xn). 

33,  p.  58,  the  equation  of  PqT^  may  be  written  in 


=^1 

dxjo 


(x  -  Xo). 


A=^\;    likewise  B=^']- 
Thus  the  equation  of  the  tangent  plane  is 


^o)! 


or,  what  is  the  same  thing, 

(6)  z  -Zo=  -^     (x-Xo)  +  ~\  (y-  yo)- 


322  SEVERAL  VARIABLES  [IX,  §  164 

It  is  important  to  notice  the  great  similarity  between  this 
equation  and  the  equation 

of  §  148.  In  fact  (7)  expresses  the  fact  that  if  dx,  dy  are 
measured  parallel  to  the  x  and  y  axes  from  the  point  of  tan- 
gency  {x^,  y^,  Zq),  dz  represents  the  height  of  the  tangent  plane 
above  (xq,  yo,  Zq).  Equation  (7)  furnishes  a  good  means  of  re- 
membering (6). 

165.  Extremes  on  a  Surface.  If  a  function  z=f(x,  y)  is 
represented  geometrically  by  a  surface,  it  is  evident  that 
the  extreme  values  of  z  are  represented  by  the  points  on  the 
surface  which  are  the  highest,  or  the  lowest,  points  in  their 
neighborhood : 

(1)  f{x^,  yo)  >f{xo  +  h,ijo  +  k),  if  / {xo,  yo)  is  a  maximum, 

(2)  /(.To,  yo)  <f(xo  +  h,yo  +  k),  if  /  {Xo,  yo)  is  a  minimuvi, 

for  all  values  of  h  and  k  for  which  /r  +  k^  is  not  zero  and  is  not 
too  large. 

It  is  evident  directly  from  the  geometry  of  the  figure  that 
the  tangent  plane  at  such  a  j^oint  is  horizontal. 

This  results  also,  however,  from  the  fact  that  the  section  of  the  surface 
by  the  plane  x  —  Xo  must  have  an  extreme  at  (x^,,  yo) ;  hence  [df/dij^Q, 
which  is  the  slope  of  this  section  at  (xq,  Vo),  must  be  zero  ;  likewise 
\_df/dx']o,  the  slope  of  the  section  through  (Xq,  i/q)  by  the  plane  y  =  J/,,, 
must  be  zero.  Hence  equation  (5),  §  164,  reduces  to  z  —  Zq  =  0,  which 
is  a  horizontal  plane. 

A  point  at  which  the  tangent  plane  is  horizontal  is  called  a 
critical  point  on  the  surface.  The  following  cases  may  present 
themselves. 

(1)  TJie  surface  may  c\d  through  its  tangent  plane;  then  there 
is  no  extreme  at  (x'o,  y^. 


IX,  §165]       TANGENT  PLANE  — EXTREMES  323 

This  is  what  happens  at  a  point  on  a  surface  of  the  saddleback  type 
shown  by  a  hyperbolic  paraboloid  at  the  origin  ;  a  homelier  example  is  the 
depression  between  the  knuckles  of  a  clenched  fist. 

(2)  The  surface  may  just  touch  its  tangent  plane  along  a  whole 
line,  hut  not  j)ierce  through;  then  there  is  ivhat  is  often  called  a 
weak  extreme  at  {xq,  y^)  ;  that  is,  z  =f{x,  y)  lias  the  same  vialue 
along  a  whole  line  that  it  has  at  (.Tq,  y^),  but  otherwise /(a;,  y) 
is  less  than  [or  greater  than]  /{xq,  y^). 

This  is  what  happens  on  the  top  of  a  surface  which  has  a  rim,  such  as 
the  upper  edge  of  a  water  glass,  or  the  highest  points  of  an  anchor  ring 
lying  on  its  side.  Most  objects  intended  to  stand  on  a  table  are  provided 
with  a  rim  on  which  to  sit ;  they  touch  the  table  all  along  this  rim,  but  do 
not  pierce  through  the  table. 

(3)  The  surface  may  touch  its  tangent  playie  only  at  the  point 
(a^o,  2/0)/  t^^^^^  ^  =f{^i  y)  ^^  «'i  extreme  at  {xq,  y^) :  a  minimum, 
if  the  surface  is  wholly  above  the  tangent  plane  near  (xq,  y^ ; 
a  maximum,  if  the  surface  is  wholly  below. 

The  shape  of  the  clenched  fist  gives  many  good  illustrations  of  this  type 
also.    Examples  of  formal  algebraic  character  occur  below. 

Example  1.    For  the  elliptic  paraboloid  2  =  x^  +  t/2  the  tangent  plane  at 

2  -  ^0  =  2  Xo  (a;  -  Xo)  +  2  j/o  (y  -  Vq)  , 

which  is  horizontal  if  2  Xq  =  2  »/o  =  0  ;  this  gives  Xq  =  y^  =  2o  =  0»  hence 
(x  =  0,  2/  =  0)  is  the  only  critical  point. 

At  (x  =  0,  y  =  0),  2  has  the  value  0  ;  for  any  other  values  of  x  and  y, 
z  (  =  x2  -\-  y-)  is  surely  positive.  It  follows  that  2  is  a  minimum  at  x  =  0, 
2/  =  0. 

Example  2.  In  experiments  with  a  pulley  block  the  weight  10  to  be 
lifted  and  the  pull  p  necessary  to  lift  it  were  found  in  three  trials  to  be 
(in  pounds)  (pi  =  5,  toi  =  20),  (p.>  =  9,  ioo  =  bO),  (p.^^lo,  703  =  90). 
A.ssuming  that  p  =  aw  +  /3,  find  the  values  of  a  and  /3  which  make  the 
sum  S  of  the  squares  of  the  errors  least.  (Compare  Ex.  18,  p.  69,  and 
§  121,  p.  229.) 

Computing  p  by  the  formula  aio+p,  the  three  values  are  p'i  =  20a  +  p, 
p'2  =  50  a  +  /3,  p'3  —  90  a  +  p.      Hence  the  sum   of  the  squares  of  the 


324  SEVERAL   VARIABLES  [IX,  §  165 

errors  is 

S={p'i-  Pi)  2  +  (p'2  -  P2Y  +  (P'z  -  PzY 

=  (20  a  +  ^  -  5)2  +  (50  a  +  iS  -  9)2  4-  (90  a  +  /3  -  15)2. 

In  order  that  ^S  be  a  minimum,  we  must  have 

1  :^'  =  20  (20  «  +  iS  -5)  +  50  (50  a  +  /3  -  9)  +  90(90  a  +  /3-  15)  =  0. 

1  £i?  =  (20  «  +  ^  -  5)  +  (50  a  +  ^  -9)  +  (90  «  +  ^  -  15)  =  0. 

that  is,  after  reduction, 

1100  a  +  16  ^  -  190  =  0,  «  =  lU  =  .143, 

whence 
160  «  +  3  ^  -  29  =  0,  /3  =  -V?if  =  2-03. 

If  the  usual  grapli  of  the  values  of  p  and  w  is  drawn,  it  will  be  seen  that 
p  =  aw  +  /3  represents  these  values  very  well  for  a  =  .143,  j3  =  2.03  and  it 
is  evident  from  the  geometry  of  the  figure  that  these  values  render  S  a 
minimum,  S  =  .0545  ;  for  any  considerable  mcrease  in  either  «  or  /3  veiy 
evidently  makes  >S'  increase.  Since  this  is  the  only  critical  point,  it 
surely  corresponds  to  a  minimum,  for  the  function  S  has  no  singularities. 

This  conclusion  can  also  be  reached  by  thinking  of  ^S"  as  represented 
by  the  heights  of  a  surface  over  an  a^  plane,  and  considering  the  section 
of  that  surface  by  the  tangent  plane  at  the  point  just  found  as  in  Ex.  3 
below  ;  but  in  this  problem  the  preceding  argument  is  simpler. 

It  IS  customary  to  assume  that  the  values  of  a  and  /3  which  make  S  a 
Ininimum  are  the  best  compromise,  or  the  "most  probable  values "  ; 
hence  the  most  probable  formula  for  p  is  p  —  .143  lo  -\-  2.03. 

The  work  based  on  more  than  three  trials  is  quite  similar ;  the  only 
change  being  that  S  has  n  terms  instead  of  3  if  n  trials  are  made. 

Example  3.  Find  the  most  economical  dimensions  for  a  rectangular 
bin  with  an  open  top  which  is  to  hold  500  cu.  ft.  of  grain. 

Let  X,  ?/,  h  represent  the  width,  length,  and  height  of  the  bin,  respec- 
tively. Then  the  volume  is  xyh  ;  hence  xyh  =  500  ;  and  the  total  area  z 
of  the  sides  and  bottom  is 

(a)  z  =  xy  +  2ky  +  2hx^xy  +  '-^+m. 

X  y 

If  this  area  (which  represents  the  amount  of  material  used)  is  to  be  a 
minimum,  we  must  have 

^  ^  dx      ^        x^         '    dy  y2 


The  plane 

2  =  300 


—15— 
Fig.  71 


IX,  §166]        TANGENT  PLANE  — EXTREMES  325 

Substituting  from  the  first  of  these  the  value  y  =  1000/x-  in  the  second, 
we  find 

r^  y 

(c)  X —  0,  whence  x  =:  0,  or  x  =  10. 

1000  ,„. 

The  value  a;  =  0  is  obviously  not  worthy  of  any 
consideration  ;  the  value  x  —  10  gives  y  =  1000/x-     — q- 
=  10  and  h  =  bOO/(xy)  =  o. 

The  value  of  z  when  x  =  10,  y  =  10  is  300.  If 
the  equation  (a)  is  represented  graphically  by  a  surface,  the  values  of  z 
being  drawn  vertical,  the  section  of  the  surface  by  the  plane  z  =  300  is 
represented  by  the  equation 

(d)  xy  +  i^  +  ^^  =  300,  or  x^  -  300  xy  +  1000(x  +  y)=0. 

X  y 

This  equation  is  of  course  satisfied  by  x  =  10,  y  =  10.  If  we  attempt  to 
plot  the  curve  near  (10,  10),  — for  example,  if  we  set  ?/  =  10  +  A;  and  try 
to  solve  for  x  in  the  resulting  equation  : 

(10  +  kyx"-  -  (300  k  +  2000)x  +  1000(10  +  k)  =  0, 

the  usual  rule  for  imaginary  roots  of  any  quadratic  ax^  +  6x  +  c  =  0 
shows  that 

62  -  4  ac  =-  1000  k^-iAk-^s-  30]  <0 

for  all  values  of  k  greater  than  —  7.5.  Hence  it  is  impossible  to  find  any 
other  point  on  the  curve  near  (10,  10).  It  follows  that  the  horizontal 
tangent  plane  z  —  300  cuts  the  surface  in  a  single  point ;  hence  the  sur- 
face lies  entirely  on  one  side  of  that  tangent  plane.  Trial  of  any  one 
convenient  pair  of  values  of  x  and  y  near  (10,  10)  shows  that  z  is  greater 
near  (10,  10)  than  at  (10,  10)  ;  hence  the  area  z  is  a  minimum  when 
X  =  10,  2/  =  10,  which  gives  h  =<>. 

166.  Final  Tests.  Final  tests  to  determine  whether  a  func- 
tion fix,  y)  has  a  maximum  or  a  minimum  or  neither,  are 
somewhat  difficult  to  obtain  in  reliable  form.  Comparatively 
simple  and  natural  examples  are  known  which  escape  all  set 
rules  of  an  elementary  nature.*     (See  Example  1  below.) 

*  For  a  detailed  discussion,  see  Goursat-Hedrick,  Mathematical  Analysis, 
Vol.  I,  p.  118. 


326 


SEVERAL  VARIABLES 


[IX,  §  166 


One  elementary  fact  is  often  useful:  if  the  surface  has  a 
maximum  at  (xq,  y^),  every  vertical  section  through  (xq,  y^  has  a 
maximum  there.  Thus  any  critical  point  (xq,  y^)  may  be  dis- 
carded if  the  section  by  the  plane  x  =  Xq  has  no  extreme  at 
that  point,  or  if  it  has  the  opposite  sort  of  extreme  to  the 
section  made  hy  y  =  y^. 

The  safest  final  test,  and  the  one  very  easy  to  apply,  is  to 
actually  draw  the  section  of  the  surface  made  by  the  horizontal 
tangent  plane,  as  in  Ex.  3,  §  165.  Then  a  test  of  a  few  values 
quickly  settles  the  matter. 


Example   1.     The  surface   z  =  (y  —  x!^)  (y  —  2  x'^)    has    critical   points 

where 

¥.=  -exy +  8x^  =  0.    S^  =  22/-3x2  =  0; 
dx  '     dy 

that  is,  the  only  critical  point  is  (x  =  0,  y  =  0).     The  tangent  plane  at 
that  point  is  2  =  0.     This  tangent  plane  cuts  the  surface  where 

(y-x^){y-2x'')=0; 

that  is,  along  the  two  parabolas 
y  =  x^,y-2  x"^.  At  a;  =  0,  2/  =  1, 
the  value  of  2  is  +  1 ;  hence  z  is 
positive  for  points  (x,  y)  inside 
the  parabola  y  =  2  x^.  At  x  =  1, 
y  =0,  the  value  of  2  is  +2  ;  hence 
z  is  positive  for  all  points  (x,  y) 
outside  the  parabola  y  =  x^.  At 
the  point  x  =  1,  ?/  =  1.5,  the  value 
of  0  is  —  .25 ;  hence  z  is  negative 
between  the  two  parabolas.  It  is 
evident,  therefore,  that  z  has  no 
extreme  at  x  =  0,  ?/  =  0. 


Fig.  72 


A  qualitative  model  of  this  extremely  interesting  surface  can  be  made 
quickly  by  molding  putty  or  plaster  of  paris  in  elevations  in  the  unshaded 
regions  indicated  above,  with  a  depression  in  the  shaded  portion. 

Another  interesting  fact  is  that  every  vertical  section  of  this  surface 
through  (0,  0)  has  a  minimum  at  (0,  0)  ;  this  fact  shows  that  the  rule 
about  vertical  sections  stated  above  cannot  be  reversed.  Moreover,  this 
surface  eludes  every  other  known  elementary  test  except  that  used  above. 


IX,  §  166]        TANGENT  PLANE  — EXTREMES  327 


EXERCISES  LXVIL  — TANGENT  PLANES  EXTREMES 

1.  Find  the  equation  of  the  tangent  plane  to  eacli  of  the  following 
surfaces  at  the  point  specified  : 

(a)  z  =  x-^  +  dy^,  (2,  1,  13).  Ans.  3  =  4x  +  18y-13. 

(b)  e  =  2x2 -4  2/2,  (3,2,  2).  Ans.  z=12x-16y-2. 

(c)  z  =  xy,  (2,  -  3,  -  6).  Ans.  3x~2y  +  z  =  6. 

(d)  z  =  (x  +  yy,  (1,  1,4).  Ans.  4x  +  4y-  z  =  4. 

(e)  z  =  23-2/2  +  2/3^  (2,  0,  0).  Ans.z^O. 

2.  The  straight  line  perpendicular  to  the  tangent  plane  at  its  point  of 
tangency  is  called  the  normal  to  the  surface. 

Find  the  normal  to  each  of  the  surfaces  in  Ex.  1,  at  the  point 
specified. 

3.  At  what  angle  does  the  plane  x  +  2y  —  z  +  S  =0  cut  the  parabo- 
loid X-  +  2/-  =  4  2  at  the  point  (6,  8,  25)  ? 

4.  Find  the  angle  between  the  surfaces  of  P^xs.  1  (a)  and  1  (6)  at  the 
point  (VlT^,  1,  22). 

Find  the  angle  between  each  pair  of  surfaces  in  Ex.  1,  at  some  one  of 
their  points  of  intersection,  if  they  intersect. 

5.  Find  the  tangent  plane  to  the  sphere  o:-  +  ?/"  +  z'^  =  25  at  the  point 
(3,  4,  0)  ;  at  (2,  4,  V5). 

6.  At  what  angles  does  the  line  x  =  2y  =  3z  cut  the  paraboloid 
y  =  x^  +  z"'. 

7.  Find  a  point  at  which  the  tangent  plane  to  the  surface  1  (a)  is 
horizontal. 

Draw  the  contour  lines  of  the  surface  near  that  point  and  show 
whether  the  point  is  a  minimum  or  a  maximum  or  neither. 

8.  Proceed  as  in  Ex.  7  for  each  of  the  surfaces  of  Ex.  1,  and  verify 
the  following  facts : 

(6)  Horizontal  tangent  plane  at  (0,  0)  ;  no  extreme. 

(c)  Horizontal  tangent  plane  at  (0,  0)  ;  no  extreme. 

(rf)  Horizontal  tangent  plane  at  every  point  on  the  line  x  +  y  =  0 ; 
weak  minimum  at  each  point. 

(e)  Horizontal  tangent  plane  at  every  point  where  2/  =  0 ;  no  extreme 
at  any  point. 


328  SEVERAL  VARIABLES  [IX,  §  166 

9.  Find  the  extremes,  if  any,  on  each  of  the  following  surfaces : 

(a)  3  =  X"-  +  4  2/2  -  4 x.     (Minimum  at  (2,  0,  —  4).) 

(&)  z^x^-Zx-y'^.     (See  Tables,  Fig.  Ij.) 

(c)  2  =  x3  -  3  X  +  y-  (x  -  4).     (See  Tables,  Fig.  Ij.) 

(d)  z=[(x-  ay  +  ?/-]  [(x  +  a)2  +  ?/2].     (Similar  to  TaftZes,  Fig.  I7. ) 

(e)  2  =  x*  —  6  X  —  2/2.     (Draw  auxiliary  curve  as  for  Fig.  Ii.) 
(/)  0  =  x^  —  4  2/2  +  xy-.     (Draw  auxiliary  curve  as  for  Fig.  I2.) 
(gr)  2  —  x^  +  2/^  —  3  xy.     (Draw  by  rotating  xy-p\ane  through  7r/4.) 

10.  Redetermine  the  values  of  a  and  /3  in  Example  2,  §  165,  if  the 
additional  information  (p  =  23,  ta  =  135)  is  given. 

11.  Find  the  values  of  u  and  v  for  which  the  expression  (aiu  +  biv 
—  ci)2  +  (aou  +  62V  —  C2)2  +  (asM  +  63U  —  C3)2  becomes  a  minimum. 
(Compare  Ex.  10. ) 

12.  Show  that  the  most  economical  rectangular  covered  box  is  cubical. 

13.  Show  that  the  rectangular  parallelopiped  of  greatest  volume  that 
can  be  inscribed  in  a  sphere  is  a  cube, 

[Hint.  The  equation  of  the  sphere  is  x"^  +  y^  +  z^  =  1  ;  one  corner  of 
the  parallelopiped  is  at  (x,  y,  z);  thenF  =  8x2/2,  where  z  =Vl— x^— 2/2  ] 

14.  Show  that  the  greatest  rectangular  parallelopiped  which  can 
be  inscribed  in  an  ellipsoid  x''/a2  +  y'-/b'-  +  z'^/c'^  =  1  has  a  volume 
F=  8  a6c/ (3V3). 

15.  The  points  (2,  4),  (6,  7),  (10,  9)  do  not  lie  on  a  straight  line. 
Under  the  assumptions  of  Ex.  2,  §  165,  show  that  the  best  compromise 
for  a  straight  line  which  is  experimentally  determined  by  these  values  is 
24  2/  =  15  X  +  70. 

16.  The  linear  extension  E  (in  inches)  of  a  copper  wire  stretched  by  a 
load  W  (in  pounds)  was  found  by  experiment  (Gibson)  to  be  (  PT  =  10, 
E  =  .06),  (  TT  =  30,  ^  =  .17),  (  >r  =  60,  ^  =  .32).  Find  values  of  a  and 
/3  in  the  formula  E  =  aW  +  ^  under  the  assumptions  of  §  165. 

17.  The  readings  of  a  standard  gas  meter  8  and  that  of  a  meter  T 
being  tested  were  found  to  be  (  T  =  4300,  S  =  500),  (T  =  4390,  S  =  600), 
(T'  =  4475,  6' =  700).  Find  the  most  probable  values  in  the  equation 
T  =  aS  +  )3  and  explain  the  meaning  of  a  and  of  /3. 


IX,  §  167]  TANGENTS  AND  NORMALS  329 

18.  The  temperatures  d°  C.  at  a  depth  d  in  feet  below  the  surface  of  the 
grouud  iu  a  iiiiue  were  found  to  be  d  =  100  ft.,  6  =  15^.7,  d  =  200  ft., 
0  —  Iti'^.d,  (1  —  300  ft.,  6  =  17^.4.  Find  an  expression  for  the  temperature 
at  any  depth. 

19.  Redetermine,  under  the  assumptions  of  §  166,  the  most  probable 
values  of  the  constants  iu  Exs.  1-5,  p.  2ot). 

20.  The  points  (10,  3.1),  (3.3,  1.6),  (1.25,  .7)  lie  very  nearly  on  a 
curve  of  the  form  «/x  +  ^/y  =  1.  Use  the  reciprocals  of  the  given 
values  to  find  the  most  probable  values  of  «  and  /3. 

21.  The  sizes  of  boiler  flues  and  pressures  under  which  they  collapsed 
were  found  by  Clark  to  be  {d  =  30,  p  =  76),  (d  =  40,  p  =  45),  (d  =  50, 
p  =  30).  These  values  satisfy  very  nearly  an  equation  of  the  form 
p  z=  k  •  d"  or  logp  =  n  log  d  +  log  k,  where  d  is  the  diameter  in  inches, 
and  p  is  the  pressure  in  pounds  per  square  inch.  Using  the  logarithms 
of  the  given  numbers,  find  the  most  probable  values  for  n  and  log  k. 

22.  Recompute,  under  the  assumptions  of  §  165,  as  in  Ex.  21,  the 
values  of  constants  in  Exs.  17,  19,  pp.  232-233. 

167.  Tangent  Planes.  Implicit  Forms.  If  the  equation  of 
a  surface  is  given  in  imjilicit  form,  F{x,  y,  z)  —  0,  taking  the 
total  differential  we  find  : 

/ix  dF.     ,  dF  J     ,  dF,        -. 

(1)  dx  +  -—dy  +  -~dz  =  0. 
ox  ay  dz 

But,  by  virtue  of  F(x,  y,  z)  =  0,  any  one  of  the  variables,  say 
z,  is  a  function  of  the  other  two ;  hence 

(2)  dz=^dx  +  ^^dy. 

dx  ay 

Putting  this  in  the  total  differential  above  and  rearranging: 

^  ^  \dx       dz  dx)      ^\dy^  dz  dyj  ^ 

But  dx  and  dy  are  independent  arbitrary  increments  of  x  and 
of  y  J  and  since  the  equation  is  to  hold  for  all  their  possible 


330  SEVERAL  VARIABLES  [IX,  §  167 

pairs   of  values,  the  coefficients  of   dx   and   dy  must   vanish 
separately.     This  gives 

.^.  dz  ^      dF/dx      dz  ^      dF/dy 

^  ^  dx         dF/dz'    dy~     dFjdz 

Substituting  these  values  in  the  equation  of  the  tangent  plane, 
and  clearing  of  fractions,  we  obtain 


X 


(6) 


fl/^-^'^  +  fl'^-^'^  +  flo^^--^^"' 


the  equation  of  the  tangent  plane  at  {x^,  y^,  Zq)  to  the  surface 
F{x,y,z)=0. 

168.  Line  Normal  to  a  Surface.  The  direction  cosines  of 
the  tangent  plane  to  a  surface  whose  equation  is  given  in  the 
explicit  form  z  =f(x,  y)  are  proportional  (§  164)  to 

(1)  dz/dx^,    dz/dy\,  and  -  1. 

Hence  the  equations  of  the  normal  at  {xq,  y^,  Zq)  are 


(2) 


2/o       z-Zo 


dz/dx^o      dz/dy^o        —  1 


The  direction  cosines  of  a  surface  whose  equation  is  given  in 
the  implicit  form  F(x,  y,  z)  =0  are  proportional  to 

(3)  dF/dx-]o,    dF/dy-],,    dF/dz],, 

so  that  the  equations  of  the  normal  to  this  surface  are 

x-Xq        y  —  yo        ^-zo 


(4) 


dF/dx]o      dF/dy]o      dF/dz], 


169.  Parametric  Forms  of  Equations.  A  surface  S  may 
also  be  represented  by  expressing  the  coordinates  of  any  point 
on  it  in  terms  of  two  auxiliary  variables  or  parameters : 

\_S]  X  =f(u,  v),   y  =  (f>(u,  v),   z  =  ij/{u,  V). 


IX,  §  169]  TANGENTS  AND  NORMALS  331 

If  we  eliminate  u  and  v  between  these  equations,  we  obtain  the 
equation  of  the  surface  in  the  form  F  (x,  y,  z)  =  0. 

Similarly  a  curve  C  may  be  represented  by  giving  x,  y,  z  in 
terms  of  a  single  auxiliary  variable  or  parameter  t : 

[C]  x^fit),   y=<t>(t),   z  =  ^(t). 

The  elimination  of  t  from  each  of  two  pairs  of  these  equations 
gives  the  equations  of  two  surfaces  on  each  of  which  the  curve 
lies,  in  the  form  (4),  §  163.  In  particular,  taking  t  =  x  gives 
the  curve  as  the  intersection  of  the  projecting  cylinders : 

[P]  y  =  <f>(x),      z  =  ^{x). 

If,  in  the  parametric  equations  of  a  surface,  one  parameter  (say  u)  is 
kept  fixed  while  the  other  varies,  a  space-curve  is  described  which  lies  on 
the  surface.  Now  if  u  varies,  this  curve  varies  as  a  whole  and  describes 
the  surface.  The  curve  on  which  u  keeps  the  value  k  is  called  the  curve 
u  =  k.  Similarly,  keeping  v  fixed  while  ?<  varies  gives  a  curve  v  =  k' . 
The  intersection  of  an  u  =  k  with  an  v  =  k'  gives  one  or  more  points 
(k,  k')  on  the  surface.  The  pair  of  numbers  (^•,  k')  are  called  the  curvi- 
linear coordinates  of  points  on  the  surface. 

Simple  examples  of  such  coordinates  are  the  ordinaiy  rectangular 
coordinate  system  and  the  polar  coordinate  system  in  a  plane.  Thus 
(2,  3)  means  the  point  at  the  intersection  of  the  lines  a:  =  2,  ?/  =  3  of  the 
plane  ;  in  polar  coordinates,  (5,  30°)  means  the  point  at  the  intersection 
of  the  circle  r  =  5  with  the  line  0  =  30°. 

Example  1.  The  equations  of  the  plane  x  +  y  -{-  z  =  1  may  be  written, 
in  the  parametric  form  : 

x=  u,         y  =  V,        z  z=l  —  u  —  v. 

Let  the  student  draw  a  figure  from  these  equations  by  inserting  arbitrary 
values  of  u  and  v  and  finding  associated  values  of  x,  y,  z.  Another  set  of 
parameter  equations  which  represent  the  same  plane  is 

x  =  u  +  v,         y  =  u  —  v,         z  =  0  —  2u  +  I. 

Thus  several  different  sets  of  parameter  equations  may  represent  the 
same  surface. 

In  the  first  form,  put  u  =  k.  Then,  as  v  varies,  we  obtain  the  straight 
line 

z  =  k,        y  =  v,        z  —  \  —  k  —  V, 


332  SEVERAL  VARIABLES  [IX,  §  169 

which  lies  in  the  given  plane.  As  k  varies  this  line  varies ;  its  different 
positions  map  out  the  entire  plane.  Likewise,  u  =  A;'  is  a  line  varying 
with  k'  and  describing  the  plane.  The  intersection  of  two  of  these  lines, 
one  from  each  system,  is  point  {k,  k')  of  the  plane. 

Example  2.     The  sphere  x^  +  t/^  +  ^2  -  cfl  may  be  represented  by  the 
equations : 

X  =  a  cos  9  cos  0,        y  =  a  cos  ^  sin  0,        z  =a  sin  6. 

Here  the  parameters  6  and  (p  are  respectively  the  latitude  and  the  longi- 
tude. Thus  0  =  k  is  a  parallel  of  latitude  ;  <^  =  ^•'  is  a  meridian  ;  and 
their  intersection  (A-,  k')  is  a  point  of  latitude  k  and  longitude  k' .  [If  a 
is  allowed  to  vary,  the  equations  of  this  example  define  polar  coordinates 
in  space  ;  but  the  colatitude  90°  —  6  is  often  used  in  place  of  ^.] 

Example  3.     The  equations 

X  =  a  cos  t,        y  =  a  sin  t,        z  =  bt, 

represent  a  space  curve,  namely  a  helix  drawn  on  a  cylinder  of  radius  a 
with  its  axis  along  the  ^-axis.  The  total  rise  of  the  curve  during  each 
revolution  is  2  nb. 

If  a  is  replaced  by  a  variable  parameter  u,  the  helix  varies  with  u,  and 
describes  the  surface 

X  =  u  cos  t,         y  =  u  sin  ^,         z  =  bt, 

which  is  called  a  helicoid.  The  blade  of  a  propeller  screw  is  a  piece  of 
such  a  surface. 

170.   Tangent  Planes  and  Normals.    Parameter  Forms. 

When  a  surface  is  given  by  means  of  parametric  equations, 

(1)  x=/(m,  v),         y  =  4>(u,v),         z  =  ^{u,v), 

the  equation  of  the  tangent  plane  is  found  as  follows.  Elimination  of  u 
and  V  would  give  the  equation  in  the  implicit  form  jP(x,  ?/,  z)  -  0.  If  the 
parametric  values  of  x,  y,  z  are  substituted  in  this  equation  the  resulting 
equation  is  identically  true,  since  it  must  hold  for  all  values  of  the  inde- 
pendent parameters  ii,  v;  hence 

(2)  ^  =  0,  and  ^  =  0, 
^  du  dv 

that  is  4 

dx  du      dy  du      dz  du        '    dx  dv      dy  dv      dz  dv 


IX,  §  170]  TANGENTS  AND  NORMALS 

Solving  these,  we  find : 


333 


(4) 


dx  '  dy  '  dz 


dy    8z 

dz 

dx 

dz    dy_ 

du    du 

du 

du 

du    du 

dy    dz 

dz 

dx 

dx    dy 

dv    cv 

dv 

dv 

dv    dv 

hence  the  equation  of  the  tangent  plane  is 


(x  —  Xo) 


dy^  dz^ 

du  du 

dy  dz 

dv  dv 


+  (y  -  2/o) 


az  dx 

du  du 

dz  dx 

dv  dv 


+  (^ 


dx 

dy 

ru 

du 

zo) 

dx 

dy 

dv 

dv 

while  the  equations  of  the  normal  are 
X  —  .To  y  —  2/0 


di  dz_ 

du    du 

cz    ex 
du    du 

dx  dy_ 
du    du 

dy    dz 
dv    dv 

0 

dz    dx 
dv    dv 

1, 

dx    dy_ 

dv    dv 

=  0; 


EXERCISES  LXVra.  -  EQUATIONS   NOT   IN  EXPLICIT  FORM 

1.  Determine  the  tangent  plane  and  the  normal  to  the  ellipsoid 
x-^  +  4?/2  +  2-  =  36  at  the  point  (4,  2,  2),  first  by  solving  for  z,  by  the 
methods  of  §  164  ;  then,  vfithout  solving  for  z,  by  the  methods  of  §§  167-168. 

2.  Detei-mine  the  tangent  planes  and  the  normals  to  each  of  the  fol- 
lowing surfaces,  at  the  points  specified  : 

(a)  x^+y'^  +  z^  =  a2  at  (xq,  t/o-  2^0). 
(6)  x^  -  4  ?/2  +  ^2  =  36  at  (6,  1,  2). 

(c)  x2  -  4y2  _  9^2  =  36  at  (7,  1,  1). 

(d)  x2  +2/2  _  «2  =  0  at  (3,  4,  5). 

(e)  a^  +  a:2y  -  2  22  =  0  at  (1,  1,  -  1). 
(/)  22  =  gx+y  at  (0,  0,  1). 

3.  Find  the  angle  between  the  tangent  planes  to  the  ellipsoid  4  x2  4-9  j/2 
+  36  ^2  =  36  at  the  points  (2,  1,  z^)  and  (-  1,  -  1.  21). 

4.  At  what  angle  does  the  2-axis  cut  the  surface  z^  =  e^+i'  ? 


334  SEVERAL  VARIABLES  [IX,  §  170 

5.  Obtain  the  equation  of  the  tangent  plane  to  the  helicoid 

X  =  u  cos  V,         y  =  n  sin  V,        z  —  v, 
at  the  point  m  =  1,  u  =  7r/4. 

6.  Taking  the  equations  of  a  sphere  in  terms  of  the  latitude  and  longi- 
tude (Example  2,  §  169),  find  the  equation  of  its  tangent  plane  and  the 
equations  of  the  normal  at  a  point  where  6  =  <t>  —  45° ;  at  a  point 
where  0  =  60°,  <p  =  30^. 

7.  Eliminate  u  and  v  from  the  equations  x  =  u  +  v,  y  =  u  —  v,  z=:  nv, 
to  obtain  an  equation  in  x,  y,  and  z.  Find  the  equation  of  the  tangent 
plane  at  a  point  where  w  =  3,  v  =  2,  by  the  methods  of  §  164  ;  then  by  the 
methods  of  §  170  directly  from  the  given  equations. 

8.  Write  the  equation  of  the  tangent  plane  to  the  surface  used  in  Ex. 
7  at  any  point  (Xq,  y^,  Zq).  At  what  point  is  the  tangent  plane  hori- 
zontal ?    Is  z  an  extreme  at  that  point  ? 

9.  Proceed  as  in  Ex.  7  for  each  of  the  following  surfaces  : 
(a)  X  =  r  cos  0,   y  =  rsiaO,   z  =  r,  at  r  =  2,  6  =  7r/4. 

U  +  V  U  +  V  it  +  V 

(c)  X  =:  —  S  u  +  2  V,   y  =2u—  V,   z  =  €"■+",  at  (uq,  Vq)  . 

(d)  X  =  2  cos  0  cos  <!>,   y  =  S  cos  0  sin  4>,   z  =  sin  ^,  at  ^  =  <^  =  7r/4. 

10.  The  surfaces  z  =  x'^  —  4  y^  and  z  =  6x  intersect  in  a  curve,  whose 
equations  are  the  two  given  equations.  Find  the  tangent  line  to  this 
curve  at  the  point  (8,  2,  48)  by  first  finding  the  tangent  planes  to  each  of 
the  surfaces  at  that  point ;  the  line  of  intersection  of  these  planes  is  the 
required  line. 

11.  Find  the  tangent  line  to  the  curve  defined  by  the  two  equations 
16  x2  -  3  2/2  =  4  2  and  9  x2  +  3  2/2  _  22  -  20  at  (1,  2,  1). 

171.   Area  of  a  Curved  Surface.     Let  s  he  a  portion  of  a  curved 

surface  and  R  its  projection  on  the  ,T?/-plane.  In  B  take  an  element 
Ax  Ay  and  on  it  erect  a  prism  cutting  an  element  AS  out  of  S.  At  any 
point  of  AS,  draw  a  tangent  plane.  The  prism  cuts  from  this  an  ele- 
ment AA.  The  smaller  Ax  Ay  (and  therefore  A*S')  becomes,  the  more 
nearly  will  the  ratio  AA/AS  approach  unity,  since  the  limit  of  this  ratio 
isl. 

Suppose  now  that  the  area  B  is  all  divided  up  into  elements  AxAy  and 


IX,  §  171] 


AREA  OF  A   SURFACE 


335 


that  on  each  a  prism  is  erected. 
The  area  (S'  will  thus  be  divided  up 
into  elements  AS  and  there  will  be 
cut  from  the  tangent  plane  at  a  point 
of  each  an  element  A^.  One  thus 
gets 

(1)  S: 


lim  V  A^. 


But  if  7  is  the  acute  angle  that      */ 
the  normal  to  any  AA  makes  with  the 
2-axis,  we  have 

(2)  AA  =  sec  y  AxAy; 

hence 
(3) 


im  V  A^  =  lim  X  (sec  y  AxAy)  =  \   \   sec  y  dx  dy. 


:  li 

Aj=0  ■ 

Ak:^) 


Of  course  sec  y  is  a  variable  to  be  expressed  in  terms  of  x  and  y  from  the 
equation  of  the  surface.  The  limits  of  integration  to  be  inserted  are  the 
same  as  if  the  area  of  B  were  to  be  found  by  means  of  the  integral 

lldxdy. 

If  the  surface  doubles  back  on  itself,  so  that  the  projecting  prisms  cut 
it  more  than  once,  it  will  usually  be  best  to  calculate  each  piece  separately. 

When  the  equation  of  the  "surface  is  given  in  the  form  z  =f{x,  y),  the 
direction  cosines  of  the  normal  are  given  by 


cos  a  :  cos  /3  :  cos  y  =  ■ 


1. 


Taking  cos  y  positive,  that  is  y  acute,  we  may  write 
(4) 


gjnaD^'' 


and 


4 


-=nv(i)^+(i)'™»- 


The  determination  of  sec  7,  when  the  .surface  is  given  in  the  form 
F(x,  y,  z)  =  0,  is  performed  by  straightforward  transformations  similar 
to  those  used  in  §§  167-170;  they  are  left  to  the  student. 


336      .  SEVERAL  VARIABLES  [IX,  §  171 

EXERCISES  LXIX.-AREA  OF   A  SURFACE 

1.  Calculate  the  area  of  a  sphere  by  the  preceding  method. 

2.  A  square  hole  is  cut  centrally  through  a  sphere.  How  much  of 
the  spherical  surface  is  removed? 

3.  A  cylinder  intersects  a  sphere  so  that  an  element  of  the  cylinder 
coincides  with  a  diameter  of  the  sphere.  If  the  diameter  of  the  cylinder 
equals  the  radius  of  the  sphere,  what  part  of  the  spherical  surface  lies 
within  the  cylinder  ? 

4.  How  much  of  the  surface  z  =  xy  lies  within  the  cylinder  x^+y'^=l? 

5.  How  much  of  the  conical  surface  z"^  =  x"^  +  ?/2  lies  above  a  square 
in  the  x2/-plane  whose  center  is  the  origin  ? 

6.  Show  that  if  the  region  B  of  §  171  be  referred  to  ordinary  polar 
coordinates,  AA  =  rsecy  ArAO,  approximately.     (See  [B],  p.  212.) 

7.  Using  the  result  of  Ex.  6,  show  that  S  =  \  I »'  sec  y  dr  dd. 

8.  Show  that,  for  a  surface  of  revolution  formed  by  revolving  a  curve 
whose  equation  is  z  =f(x)  about  the  z  axis,  sec  7  =  Vl  +  \_df{i')/dr']--, 
where  r  =  Va;^  +  y'^. 

9.  By  means  of  Exs.  7,  8,  show  that  the  area  of  the  surface  of  revo- 
lution mentioned  in  Ex.  8  is 


where  a  is  the  value  of  r  at  the  end  of  the  arc  of  the  generating  curve. 
(See  Ex.  13,  p.  129.) 

10.  Compute  the  area  of  a  sphere  by  the  method  of  Ex.  9. 

11.  Eind  the  area  of  the  portion  of  the  paraboloid  of  revolution  formed 
by  revolving  the  curve  z^  =  2  mx  about  the  x  axis,  from  x  =  0  to  x  =  k. 

12.  Show  that  the  area  of  the  surface  of  an  ellipsoid  of  revolution 
is  2  7r&  [?) +(a/e)  sin-^e],  where  «  and  b  are  the  semiaxes  and  e  the 
eccentricity,  of  the  generating  ellipse. 

13.  Show  that  the  area  generated  by  revolving  one  arch  of  a  cycloid 
about  its  base  is  G4  ira^/S. 

14.  Show  that  the  area  of  the  surface  generated  by  revolving  the 
curve  a;2/3  -j-  2^2/3  _  (j2/3  about  one  of  the  axes  is  12  way 5. 


IX,  §172]    TANGENTS   TO   CURVES  — LENGTHS 


337 


172.   Tangent  to  a  Space  Curve.     Let  the  equation  of  the  curve 

be  given  in  paninietric  form  x  =/(<),  y  =  <p(t),  z  =  f  (<).  Let  Po  =  (a;o, 
2/0,  2o)  he  the  point  on  the  curve  where  t  —  to.  Let  §  be  a  neighboring 
point  on  the  curve  where  (  =  (o  +  A^ 

The  direction  cosines  of  the  secant  PoQ  are  proportional  to  Ax/M, 
Ay/ At,  Az/At ;  hence  its  equations  are 


(1) 


X-  xo  _y  —  yo  _z  —  zq 
Ax /At      Ay /At      Az/At 


As  At  =  0,  these  become 


(2) 


X  -  Xq    _    y  -  yo    _    Z  —  Zq 

dx/dt]o     dy/dt}o      dz/dtio 


the  equations  of  the  tangent  at 
the  point  Pq. 

If  the  curve  is  given  as  the 
intersection  of  two  projecting 
cylinders  y=f(x),  z  =  ^(x), 
we  may  join  to  these  the  third 
equation  x  =  x,  thus  conceiving  of  x, 


Fig.  74 


and  z  as  all  expressed  in  terms 


of  X.     The  equations  of  the  tangent  then  become 


(3) 


-3^0^  y-yo  _  z  -  Zq  ^ 

1  dy/dx^Q     dz/dx']o' 


If  the  curve  is  given  as  the  intersection  of  two  surfaces,  f(x,  y,  z)  =  0, 
F(x,  y,  z)  =  0,  and  if  we  think  of  x,  y,  z  as  depending  upon  a  parameter 
t,  we  find 

df_Bfdx_^^dy_^dfdz^Q 

dt     ex  dt      dy  dt      dz  dt 

and  ^=^^  +  ^'^  +  ^^=0. 

dt      dx  dt       dy  dt      cz  dt 

From  these  equations  we  obtain  dx/dt  :  dy/dt :  dz/dt,  and  we  may  write 
the  equations  of  the  tangent  at  P,,  in  the  form  : 


X  -  .r„ 

y-yo 

Z  —  Zo 

df     df 

?/.  cf_ 

i/.^ 

dy    dz 

dz   dx 

dx    dy 

BF  dF 

dF  dF 

dF  dF. 

dy    dz 

0 

dz    dx 

0 

dx    dy 

0 

338  SEVERAL  VARIABLES  [IX,  §  173 

173.    Length  of  a  Space  Curve.     The  length  of  the  chord  joining 
two  points  t  and  t  +  At  of  the  curve 


is  Ac  =  VAx^  +  A2/2  +  a^^,  or, 


^  ^  ^f  AC^       A«2  ^  A«2 

Defining  the  length  of  a  curve  between  two  points  as  the  limit  of  the  sum 
of  the  inscribed  chords  (see  §  12,  p.  18),  we  find  for  that  length: 


c^)    -isis-^cvdr+dr-d)' 


EXERCISES  LXX.  — TANGENTS  TO  CURVES  LENGTHS 

1.  Write  the  equation  of  the  tangent  at  an  arbitrary  point  of  each  of  the 
curves  in  Ex.  12,  p   320. 

2.  At  what  angle  does  a  straight  line  joining  the  earth's  South  pole 
with  a  point  in  40°  North  latitude  cut  the  40th  parallel  ? 

3.  At  what  angle  does  the  helix  x  =  2  cos  ^,  y  =  2  sin  5,  z  =  d,  cut  the 
sphere  x^  +  y'  +  z^  =  9? 

4.  Find  the  angle  of  intersection  of  the  ellipse  and  parabola  that  are 
cut  from  the  cone  z'^  =  x^  +  y^  by  the  planes  2  ^  =  1  —  x  and  z  =  1  +  x 
respectively. 

5.  Show  that  the  curves  of  intersection  of  the  three  surfaces 

2  =  y,  X2  =  2/2  +  22,  X2  +  2/2  +  ^2  =  1, 

cut  each  other  mutually  at  right  angles. 

6.  Show  the  same  for  the  curves  of  intersection  on  the  surfaces 
4x'^^9y^  +  36  z^  =  36,  3  x2  +  6  y^  -  6  ^2  =  6,  10  x2  -  15  r/2  -  6  ^2  =  30. 

7.  Calculate  the  length  of  the  curve  x  =  t,  y  =  f-,  z  =  2  «3/2,  from  t  =  0 
to  «  =  1. 

8.  Find  the  length  of  the  helix  x  =  a  cos  ^,  y  =  o  sin  5,  z  =  be,  from 
e  =  ^0  to  e  =  ^1.     What  is  the  length  of  one  turn  ? 

9.  Find  the  length  of  the  curve  x  =  sin  0,  y  =  cos  z,  from  (1,  0,  7r/2) 
to  (0,  -  1,  tt). 


IX,  §  173]  GENERAL   EXERCISES  339 

EXERCISES  LXXI— GENERAL  REVIEW      SEVERAL  VARIABLES 

[The  exercises  marked  with  an  asterisk  are  of  more  than 
usual  difficulty.  Some  of  them  contain  new  concepts  of  value 
for  which  it  is  hoped  that  time  may  be  found.  Those  of  the 
greatest  theoretical  value  are  marked  f. 

Attention  is  called  to  the  reviews  of  double  and  triple  in- 
tegration.] 

1.  Given  u  =  xy,  x  =  r  cos  6,y  =  r  sin  ^,  find  cii/cr  and  dv/c0,  first  by 
actually  expressing  n  in  terms  of  r  and  d ;  then  directly  from  the  given 
equations. 

2.  Proceed  as  in  Ex.  1  for  the  function  m  =  ta,n-'^(y/x). 

3.  Given  u  =  r-e-^,  x  =  rcose,  y  =  r  sin  e,  find  du/cx  and  du/dy,  first 
by  expressing  u  in  terms  of  x  and  y  ;  then  directly  from  the  given  equa- 
tions. 

[Hint.  In  the  second  part,  it  is  convenient  here  to  solve  the  last  two 
equations  for  r  and  0  in  terms  of  x  and  y.    But  see  Ex.  4.] 

4.*  If  X  =  r  cos  ^  and  y  =  r  sin  6,  show  by  differentiation  that 

^=l=^cos^-rsin^^,     and     ^  =  0  =  ^'sin  ^  +  r  cos  tf^. 
dx  dx  ex  ex  ex  dx 

Solve  these  equations  for  cr/dx  and  dO/dx,  and  shov?  that  du/cx  may  be 
found  in  Ex.  3  by  means  of  the  equation 

du  _dudr,dudd 
cx~drdx      dedx' 

5*  If,  in  general,  u  is  a  function  of  the  two  variables  (r,  d),  show 
that  the  last  equation  in  Ex.  4  holds  true.  Find  a  similar  equation  for 
du/dy,  and  evaluate  du/cy  in  Ex.  3  by  means  of  it. 

6.*t  If  u  is  a  function  of  any  two  variables  p  and  q,  and  if  p  and  q 
are  given  in  terms  of  x  and  y  by  two  equations  x  =f(p,  q),  y  =  <t>{p,  q), 
obtain  cu/dx  and  du/dy  by  a  process  analogous  to  that  of  Exs.  4,  5. 

7.  Proceed  as  in  Ex.  3,  by  the  methods  of  Exs.  4,  5,  in  each  of  the 
following  cases : 

(a)  u  =  r^-  cos2  6,  (b)  u  =  rc.^^         (r)  u  =  0  log  r. 

8.  Find  the  volume  of  that  portion  of  a  sphere  of  radius  4  ft,  which 
is  bounded  by  two  parallel  planes  at  distances  2  ft.  and  3  ft.,  respectively, 
from  the  center,  on  the  same  side  of  the  center. 


340  SEVERAL  VARIABLES  [IX,  §  173 

9.  Determine  the  position  of  the  center  of  mass  of  the  solid  described 
in  Ex.  8. 

10.  What  is  the  nature  of  the  field  of  integration  in  the  integral 
Show  that  the  same  integral  may  be  written  in  the  form 

lo"^'  r  -^(^'  y^  'y  ''^ + i/vJo^"^-^^^'  y^  '^  'y- 

11.  Find  the  volume  cut  from  the  sphere  x"^  +  y"^  +  z^  =  a^  by  the 
cylinder  x^  +  y^  —  ax  =  0. 

12.  Find  the  volume  cut  from  the  sphere  x'^  +  y'^  +  z-  =  a^  by  the  cone 
(x  -  ay  +  2/2  -  22  =  0. 

13.  Show  that  the  surface  of  a  zone  of  a  sphere  depends  only  upon 
the  radius  of  the  sphere  and  the  height  6  —  a  of  the  zone,  where  the 
bounding  planes  are  z  =  a  and  z  =  b. 

14.  Find  the  area  of  that  part  of  the  surface  k-z  =  xy  within  the 
cylinder  a;2  +  2/2  =  k'^. 

15.  Find  the  center  of  gravity  of  the  portion  of  the  surface  described 
in  Ex,  14,  when  k  =  I. 

16.  Find  the  moment  of  inertia  about  its  edge,  of  a  wedge  whose  cross 
section,  perpendicular  to  the  edge,  is  a  sector  of  a  circle  of  radius  1  and 
angle  30°,  if  the  length  of  the  edge  is  1,  and  the  density  is  1. 

17.  The  thrust  due  to  water  flowing  against  an  element  of  a  surface  is 
proportional  to  the  area  of  the  element  and  to  the  square  of  the  com- 
ponent of  the  speed  perpendicular  to  the  element.  Show  that  the 
total  thrust  on  a  cone  whose  axis  lies  in  the  direction  of  the  flow  is 
k7rr^vy(r^+}i^)i. 

18.  Calculate  the  total  thrust  due  to  water  flowing  against  a  segment 
of  a  paraboloid  of  revolution  whose  axis  lies  in  the  direction  of  the  flow. 
(See  Ex.  17.) 

19.  Show  that  the  thrust  due  to  water  flowing  against  a  sphere  is 
2  kirrH^/Z.  Compare  with  the  thrust  due  to  the  flow  normally  against  a 
diametral  plane  of  this  sphere. 


IX,  §  173]  GENERAL   EXERCISES  341 

20.*t   Given  a  function /(a;,  y),  consider  the  function 
(f>{t)  =/(a  +  ht,  b  +  kt), 

and  show,  by  means  of  Maclaurin's  series  for  0(f)  [see  [D*]',  §  134, 
p.  258,]  that,  upon  inserting  the  special  value  f  =  1,  we  obtain  : 

f(a  +  h,  b  +  k)  =f(a,  b)  +  \h^  +  k^~\  + 

L     ex  C'yjx=a 

+i.[h^^^+2hk-^+k^m+... '" 

2!L     dx^  dxdy  dy^Jx=a 

»=» 

(71  -  1)  1 L        c^''-i      ^         ^  ax»-2gy         J^ 

!r=b 

where  |  ^„  ]  <  ilf  (|  A  |  +  |  A;  |)"  -j-  ?(  ! ,  and  where  M  is  the  maximum  of 
the  absolute  values  of  all  the  nth  derivatives  in  a  rectangle  whose  sides 
a.Te  X  =  a,  X  =  a  +  h,  y  =  b,y  =  b  +  k.  [Taylor's  Theorem.] 

21,t  Assuming  the  truth  of  the  formula  of  Ex.  20,  show  that  the  spe- 
cial values  a  =  0,  b  =  0,  h  =  X,  k  =  y,  lead  to  the  formula 

/(.,,)=/(0,0)  +  [.|  +  !,|]_^^^^+...  +  S.. 

22.  Expand  each  of  the  following  functions  by  use  of  the  formula  of 
Ex.  21,  in  powers  of  x  and  y  as  far  as  terms  of  the  second  degree  : 

(a)  sin(x  +  j/).  (6)  e2«+3v.  (c)  cos(^x^  +  y^. 

23.  Find  the  critical  points,  if  any  exist,  for  the  surface  z  =  x-  +  2y^ 
—  4  X  —  4  y  +  10.  Is  the  value  of  z  an  extreme  at  that  point  ?  Draw  the 
contour  lines  near  the  point. 

24.  Determine  the  greatest  rectangular  parallelepiped  which  can  be 
inscribed  in  a  sphere  of  radius  a. 

25.  The  volume  of  CO2  dissolved  in  a  given  amount  of  water  at  tem- 
perature ^  is  \  e       0         6        10        15, 

[v    1.80    1.46    1.18    1.00. 
Determine  the  most  probable  relation  of  the  form  v  =  a  +  bd. 

26.  Determine  the  most  probable  relation  of  the  form  S  —  a  +  bF^ 
from  the  data :  |  P    550    650     750    850, 

\S      26      35      62      70. 

27.  Determine  the  most  probable  relation  of  the  form  y  =  ae^  from 
the  data :  f  x      1       2       3       4, 

y     .74     .27     .10     .04. 


342  SEVERAL  VARIABLES  [IX,  §  173 

28.   The  barometric  pressure  P  (inches)  at  height  H  (thousands  feet)  is 

P    30    28     26      24     22       20       18        16  , 

H    0     1.8    3.8     5.9    8.1     10.5     13.2     16.0. 

Determine  the  most  probable  values  of  the  constants  in  each  of  the 

assumed  relations  :   (a)  H -  a  +  hP  ;  (6)  H=a  +  hP  +  cP^  ;  (c)  H  = 

a  +  b  log  P  or  P  =  Ae^^.     Which  is  the  best  approximation  ? 

29t.  If  the  observed  values  of  one  quantity  y  are  iiii,  mo,  ms,  corre- 
sponding to  values  Zi,  h,  h  of  a  quantity  x  on  which  y  depends,  and  if 
y  =  ax  +  b,  show  that  the  sum 

S  =  (ah  +  b-  mi)2  +  (ah  +  b-  jno)2  +  (ah  +  b-  ws)"^ 
is  least  when 

I  h  (ah  -\-b  —  mi)  +  h  (ah  +  b  —  m-i)  +  h  (ah  +  b  —  ms)  =  0, 
I     (ah  +  b  —  Mil)  +      (ah  +  b  -  m^)  +      (ah  +b  —  mg)  =  0  ; 
that  is,  when 


a-^h^  +  b-  2^1-  V^niZi^Oanda- VZi  +  36-  ^ 


mi 


h' 


where  ^  indicates  the  sum  of  such  terms  as  that  which  follows  it. 

[Theory  of  Least  Squares.] 

30.  Show  that  the  equation  of  the  tangent  plane  to  2z  =  x^  +  y"^  at 
(xo,  2/o)  isz  +Zo  =  xxo  +  2/2/0. 

31.  Determine  the  tangent  plane  and  normal  line  to  the  hyperboloid 
a;2  _  4  2/2  +  9  ^2  ^  36  at  the  point  (2,  1,  2). 

32.  Study  the  surface  xyz  =  1.  Show  that  the  volume  included  be- 
tween any  tangent  plane  and  the  coordinate  planes  is  constant. 

33.  Study  the  surface  z  =  (x^  4-  y"^)  (x^  +  y^  -1).  Determine  the  ex- 
tremes. 

34.  At  what  angle  does  a  line  through  the  origin  and  equally  inclined 
to  the  positive  axes  cut  the  surface  2z  =  x:^  +  y^? 

35.  Determine  the  tangent  line  and  the  normal  plane  at  the  point 
(1,  3/8,  5/8)  on  the  curve  of  intersection  of  the  surfaces  x  +  y  +  z  =  2 
and  x2  +  4  2/2  -  4  22  =  0. 

36.  Determine  the  tangent  line  and  the  normal  plane  to  the  curve  x  = 
2  cost,  y  =  2  sin  t,  z  =  f^  a.t  t  =  ir/2  and  at  t  =  v. 


IX,  §  173]  GENERAL   EXERCISES  343 

37.  Find  the  length  of  one  turn  of  the  conical  spiral  x  =  t  cos  (a log  t), 
y  —  tsm  (a  log  t),  z  =  bt,  starting  from  t  =  t. 

38.  Determine  the  length  of  the  curve  x  =  a  cos  6 cos  <(>,  y  =  asin  6  cos  0, 
2  =  a  sin  0,  from  (p  =  <pi  to  <p  =  02i  where  0  is  given  in  terms  of  4>  by  the 
equation  d  =  k  log  cot  (ir/4  —  <f>/2).     (Loxodrome  on  the  sphere.) 

39.*t  Show  that  the  surfaces  /(a:,  y,  z)  =  0  and  <f>  (x,  y,  z)  =0  cut 
each  other  at  right  angles  if  /x0x  +  /y0y  +  f^^z  =  0. 

40.*  Show  that  the  surfaces 

xV(a2  +  \)  +  2/"/(&-  +  ^)  +  z-/(c-  +  X)  =  1,       a>b>c>0, 

are  always  (i)  ellipsoids  if  X>  —  c^,  (i7)    hyperboloids  of   one   sheet  if 
—  b^<\  <  —  c^,  ( Hi)  hyperboloids  of  two  sheets  if  —  a"-^ < X <  —  b-. 

(CONFOCAL    QUADRICS.) 

Show  also  that  these  surfaces  cut  each  other  mutually  at  right  angles. 

41.*t  On  the  surface  x  =f(u,  v),  y  =  <p  (u,  v),  0  =  ^  (m,  w),  show 
that    ds    (or  Vdx-  +  dy-  +  dz-)  =  VEdu^  +  2  Fdudv  +  G dv^,    where 

42,  Ux  =  r  cos  0  cos  0,  y  =  r  cos  ds'incp,  z  =  r  sin  0  (polar  coordinates), 
find  ctt/dr,  du/dd,  and  du/d(p  for  each  of  the  following  functions  : 

(a)  ?i  =  x2  +  2/2  +  2.-2,         (6)  M  =  a;2  4-  y2  _  2.2,         (c)  u  =  ze^+y. 

43.  Compute  du/dx,  du/dy,  and  g?(/cs  if  u  =  r^  (siu2  5  +  sin2  0),  where 
r,  0,  0  are  defined  as  in  Ex.  42. 

44.*  If  (r,  ^)  are  the  polar  coordinates  of  a  point  in  the  plane  whose 
rectangular  coordinates  are  (x,  y),  and  if  u  is  any  function  of  x  and  y, 
show  that 

""""ax^argx      de  cx~  cr  cd    r 

_cu  _dudr,dud6_  cu  -^  e  +  —  ^^^^ 
''~cy~drdy      dd  dy"  cr  cd     r 

45.1  Show  that  the  centroid  (x,  y)  of  a  plane  area  in  polar  coordinates 
(p,  d)  is 

r  r/)2  cos  ddpde  r  rp2  sin  ^  dp  de 

\  \pdpde  \  \pdpde 

■where  the  integrals  are  extended  over  the  given  area. 


344  SEVERAL  VARIABLES  [IX,  §  173 

46.*   Repeat  the  process  of  Ex.  44  on  the  functions  Ux  and  Uy  to  show 
that 

_  cUx  _  cux        a  _  cMj  sin  6 
'"~  dx~  cr  dd     r 

Uyy    =  2  Sin    tf    +  —f  , 

^^      dr  dd     r 

and  by  means  of  the  relations 

£^==A/£!fcos^-^5Hl^U^cos^  +  ^?nLf--^, 
dr      dr\dr  cO     r    J      df^  dd    r-        drdO 

^  =  Af^cos^-^?i^U^^cos^-£^sin»-^5l^-^^28i 
dd      dd\dr  cd     r    j      drdd  dr  dS^     r        dd     r    ' 

and  similar  relations  for  dUy/dr  and  dUy/dd,  show  that 

dH      du^      d-u       .,  a       dhi  2  sin  ^  cos  ^  ,  d'^u  sin2  d  .  du  sin^  d 

u„  =  —  =  — -  = cos-  d 1 H 

dx^       dx      or-  drdd  r  dQ^     r^        dr     r 

,  du  2  sin  6  cos  9 

^Ve        r        ' 

and 

a^M     duy      dH   -on,    d"u  2  sin  0  cos  0  ,  d'^u  cos^  d  ,  dv  cos^  d 
dy'^       dy       df^  drdd  r  dff^     r~         dr      r 

_du2sm_ecos_e 
do  r 

47.*t   By  means  of  Ex.  46,  show  that  Laplace's  Equation,  dH/dx"^ 
+  d'^u/dy'^  =  0,  is  equivalent,  in  polar  coordinates,  to  the  equation : 

3,.2      rdr      r'^ff^ 

48.   Show  that  u  =  log  r  is  a  solution  of  Laplace's  equation  (Ex.  47) 
by  direct  substitution  in  the  last  equation. 

49.*t   Complete  differentials.     If  u  =/(x,  y)  we  know  that 

du  =  {df/dx)  dx  4-  {df/dy)  dy ; 

hence  if  dji  is  given,  say  du  =  P  (x,  y)dx  +  Q  (r,  y)dy,  we  have  P  =  df/dx^ 
Q  =  df/dy.     Show  that  cP/cy  =  cQ/dx. 

50.   Given  du  =  (e'^  +  sin  ?/)  d.x  +  (pf'  +  x  cos  ?/)(f2/,  show  that  the  con- 
dition dP/dy  =dQ/dx  of  Ex,  49  is  satisfied.     [See  also  Exs.  4-11,  p.  361.] 


CHAPTER  X 

DIFFERENTIAL  EQUATIONS 

PART     I.      ORDINARY    DIFFERENTIAL    EQUATIONS     OF 
THE  FIRST  ORDER 

174.  Reversal  of  Rates.  In  Chapter  V  we  studied  the 
problem  of  finding  a  function  ichose  derivative  is  given,  i.e.  the 
problem  of  reversed  rates.     We  found  that  if 

is  given,  the  original  function  y  can  be  found  : 

at  least  to  within  an  arbitrary  constant. 

175.  Other  Reversed  Problems.  The  preceding  process  was 
applied  in  various  ways.  Thus  we  found  (§  113,  p.  206)  that 
the  distance  s  passed  over  by  a  moving  body  could  be  found  if 
the  speed  v  is  given : 

.(1)  s  =  §vdt+c;    t-  =  |=/(0. 

A  similar  process  was  applied  repeatedly  :  thus  we  found  the 
speed  V  from  the  tangential  acceleration : 

(2)  v  =  ^Jrdt  +  C;   i,  =  |  =  </»(0; 
hence 

(3)  s  =  1 1  jjrdt  +  Ci  }  at  -}-  a=^^jrdtdt  -f  C,t  +  C^ 

346 


346  DIFFERENTIAL  EQUATIONS  [X,  §  175 

Again,  in  (§  83,  p.  147),  we  found  that  {dy/dx)-r-  y  expresses 
the  relative  rate  of  change  (logarithmic  derivative);  and  we 
saw  that  the  only  function  whose  relative  rate  is  constant  is  a 
compound  interest  function : 

(4)  ^  -i-  y  =  Jc  gives  log^y  =  kx  +  c,  or  y  =  Ce**, 

where  C  =  e". 

Finally,  in  §  92,  p.  162,  we  found  that  a  damped  vibration  of 
the  form  y  =  e~"*  sin  (kx  +  e)  satisfies  a  differential  equation 
of  the  form 

(5)  ^  +  2a^  +  (fc2  +  a>  =  a 
da?  dx 


176.  Determination  of  the  Arbitrary  Constants.  The  deter- 
mination of  the  arbitrary  constants  appeared  in  the  very  first 
examples.  Thus,  in  §  54,  p.  91,  the  rate  at  which  water  is 
being  poured  into  a  tank  was  considered.  The  total  amount  y 
was  found  to  be 

y  =  r  ■  t+C, 

where  r  is  the  rate  per  second,  t  is  the  time  in  seconds,  and  C 
is  the  amount  already  in  the  tank  when  t  =  0. 

The  arbitrary  constant  C  is  determined  as  soon  as  the  value 
of  y  is  given  for  some  value  of  x.  Thus  in  the  problem  of  fall- 
ing bodies  (§  113,  p.  206),  from  the  fact  that 

jj,  ==  —  const.  =  —  g=z—  32.16  ft./sec./sec, 
we  found  that 

If  the  body  is  dropped  from  rest  (v  =  0),  at  a  height  s  =  100  ft., 
we  have 


s|     =C2  =  100,     vl     =ci  =  0, 

Je=0  J(=0 


X,  §  177]  REVERSED  RATE  PROBLEMS  347 

whence 

s=-|r/^-  +  0  +  100, 

in  which  the  arbitrary  constants  have  disappeared. 

Essentially  the  same  process  was  used  in  determining  the 
arbitrary  constants  in  a  compound  interest  law  (§  81,  p,  143). 

Finally,  in  the  case  of  direct  integration,  the  arbitrary  con- 
stant was  disposed  of  by  taking  the  difference  between  two 
values  of  y  which  correspond  to  two  given  values  of  x : 

2/]'^'=J_7''^(^)c^-«;  f=/(^),  given; 

and  the  same  scheme  was  used  in  motion  problems  (§  59, 
p.  100)  and  in  compound  interest  examples  (§  81,  p.  142). 

177.  Vital  Character  of  Inverse  Problems.  These  problems 
are  reversed  or  indirect  only  from  a  mathematical  standpoint. 
From  the  standpoint  of  science,  or  of  everyday  life,  many  such 
problems  are  more  direct  than  those  which  seem  to  be  the 
original  ones  from  a  mathematical  standpoint. 

Thus,  from  the  standpoint  of  science,  it  is  just  as  much  a 
direct  problem  to  find  the  distance  passed  over  from  a  given 
acceleration,  as  to  find  the  acceleration  from  the  distance;  as 
a  matter  of  fact  the  former  is  usually  the  real  scientific 
problem. 

,  We  found  (§  97,  p.  1G9,)  that  the  radius  of  curvature  of  any 
given  curve  is  [1  +  vi-Y'^/l',  where  m  =  chj/dXy  b  =  d-y/dx^.  If 
the  curve  is  given,  this  formula  indeed  gives  the  radius  of 
curvature.  But  it  is  more  desirable  in  practice  to  find  a  curve 
whose  radius  of  curvature  behaves  in  a  way  we  wish:  given 
the  radius  of  curvature  R  =  \p  (x),  it  is  desired  to  find  a  curve 
y  =  f(x)  which  will  actually  have  just  this  radius  at  each 
point : 


348  DIFFERENTIAL  EQUATIONS  [X,  §  177 

We  shall  solve  such  differential  equations  later  (§  191,  p.  371)  j 
just  now  it  is  important  to  see  that  they  actually  arise  in  con-  \ 
Crete  direct  scientific  and  mathematical  problems. 


178.  Elementary  Definitions.  Ordinary  Differential  Equa- 
tions. Au  ordinary  differential  equation  is  one  involving  only 
one  independent  variable.  The  derivatives  in  such  an  equa- 
tion are  therefore  ordinary  derivatives. 

An  ordinary  differential  equation  may  contain  derivatives 
of  various  orders,  and  these  derivatives  may  enter  in  various 
powers. 

The  order  of  a  differential  equation  is  the  order  of'  the 
highest  derivative  present  in  it. 

The  degree  of  a  differential  equation  is  the  exponent  of  the 
highest  power  of  the  highest  derivative,  the  equation  having 
been  made  rational  and  integral  in  the  derivatives  which  occur 
in  it. 

Thus,  equation  (5),  §  175,  is  of  the  second  order  and  first 
degree;  (1),  §  174;  (1),  §  175;  (4),  §  175,  are  of  the  first 
order  and  first  degree ;  and  (1),  §  177,  when  rationalized,  is  of 
the  second  order  and  second  degree. 


179.  Elimination  of  Constants.  Differential  equations  also 
arise  in  the  elimination  of  arbitrary  constants  from  an 
equation. 

Example  1.  Thus,  if  A  and  B  are  arbitrary  constants,  the  equation 
1/  —  Ax  +  B  represents  a  straight  line  in  the  plane,  and  by  a  proper 
choice  of  A  and  B  represents  any  line  one  pleases  in  the  plane  except  a 
vertical  line.  One  differentiation  gives  m  =  dy/dx  =  A,  which  represents 
all  lines  of  slope  A.     A  second  differentiation  gives 

(1)  flexion  =  b  =  d-y/dx-  =  0, 

which  represents  all  non-vertical  lines  in  the  plane,  since  all  these  and! 
no  other  curves  have  a  flexion  identically  zero. 


X,  §  179]  INTEGRAL  CURVES  349 

Example  2.  Any  circle  whose  radius  is  a  given  constant  r  is  repre 
sented  by  the  equation 

(2)  (x  -  A)-  +  (y  -  B)-^  =  f\ 

from  which  A  and  B  may  be  eliminated  as  in  the  preceding  example. 
Differentiating  once, 

(3)  x-A  +  {y-B)yi  =  0, 
where  y'  =  dy/dx.     Differentiating  again, 

(4)  l  +  y'-2  +  ^y_  B)y'i  =  0, 

where  y"  =  d-y/dx'^.  Solving  (3)  and  (4)  iov  x  —  A  and  y  —  B  and  sub- 
stituting these  values  into  (2),  A  and  B  are  eliminated,  giving 

(6)  (1  +  2/'2)3  =  r2t/"2. 

This  says  that  every  one  of  these  circles,  regardless  of  the  position  of  its 
center,  has  the  curvature  l/r,  —  a  statement  which  absolutely  character- 
izes these  circles. 

In  general,  if 
(6)  f{x,y,c^,c.,,—,c,)  =  (i 

is  an  equation  involving  x,  y,  and  n  independent  arbitrary  con- 
stants Ci,  C2,  •••,  c„,  n  differentiations  in  succession  with  regard 
to  X  give 

<^>         !=»'  S=»'  ••■'  '£='■' 

these  equations,  together  with  (6),  form  a  system  of  ?i-|-l 
equations  from  which  the  constants  Ci,  c.,,  •••,  c„  may  be  elimi- 
nated. The  result  is  a  differential  equation  of  the  nth  order, 
free  from  arbitrary  constants,  and  of  the  form 

(8)  <f>(x,y,y',y",  '",?/"")  =  0. 

Equation  (6)  is  called  the  primitive  or  the  general  solution 
of  (8).  The  term  general  sohition  is  used  because  it  can  be 
shown  that  all  possible  solutions  of  an  ordinary  differential 
equation  of  the  nth  order  can  be  produced  from  any  solution 
that  involves  n  independent  arbitrary  constants,  with  the  ex- 
ception of  certain  so-called  "  singular  solutions  "  not  derivable 


350 


DIFFERENTIAL  EQUATIONS 


[X,  §  179 


from  the  one  general  solution  (6)  (see  Ex.  20,  List  LXXV, 
p.  362). 

Thus,  to  solve  an  ordinary  differential  equation  of  the  nth. 
order  is  understood  to  mean  to  find  a  relation  between  the 
variables  and  n  arbitrary  constants.  These  latter  are  called 
the  constants  of  integration. 

If,  in  the  general  solution,  particular  values  are  assigned  to 
the  constants  of  integration,  a  particular  solution  of  the  dif- 
ferential equation  is  obtained. 

180.  Integral  Curves.  An  ordinary  diiferential  equation  of 
the  first  order, 

(1)  <^(^,  y,  y')  =0,    or   2/'  =f(x,  y), 

where  y'  =  dy/dx,  has  a  general  solution  involving  one  arbi- 
trary constant  c : 

(2)  F{x,y,c)=^. 

This  represents  a  singly  infinite  set  or  family  of  curves,  there 
being  in  general  one  curve  for  each  value  of  c.  Any  curve  of 
the  family  can  be  singled  out  by  as- 
signing to  c  the  proper  value. 

The  differential  equation  deter- 
mines these  curves  by  assigning,  for 
each  pair  of  values  of  x  and  y,  that 
is,  at  each  point  of  the  plane,  a 
value  of  the  slope  y'\_  =  f(x,  ?/)]  of 
the  particular  curve  going  through 
that  point.  Thus  the  curves  are 
outlined  by  the  directions  of  their 
tangents  in  much  the  way  that  iron  filings  sprinkled  over  a 
glass  plate  arrange  themselves  in  what  seem  to  the  eye  to  be 
curves  when  a  magnet  is  placed  beneath  the  glass.  Straws  on 
water  in  inotion  create  the  same  optical  illusion. 
A  differential  equation  of  the  second  order: 

<i>(x,  y,  y',  y")  =  0,    or  y"  =f(x,  y,  y'), 


X,  §  180]  INTEGRAL  CURVES  351 

has  a  general  solution  involving  two  arbitrary  constants, 

F(x,  y,  c„  Co)  =  0. 

This  represents  a  doubly  infinite  or  tioo-parameter  family  oi 
curves;  for  each  constant,  independently  of  the  other,  can 
have  any  value  whatever.  The  extension  of  these  concepts  to 
equations  of  higher  order  is  obvious. 

The  curves  which  constitute  the  solutions  are  called  the 
integral  curves  of  the  differential  equation. 

EXERCISES  LXXII.— ELIMINATION    INTEGRAL   CURVES 

Find  the  differential  equations  wliose  general  solutions  are  the  follow- 
ing, the  c's  denoting  arbitrary  constants : 

1.  x^  +  y^  =  c2.  Ans.  x  +  yy'  =  0. 

2.  x-  —  y-  =  ex.  Alls,  a;-  +  y^  =  2  xyy'. 

3.  y  =  ce'  —  |(sin  x  +  cosx).  Ans.  y'  =  y  +  sin  x. 

4.  y  =  ex  +  c^.  Ans.  y  =  y'x  +  y''^. 

5.  y  =  cx+f(c).  A71S.  y  =  y'x+f(y'). 

6.  y  =  eie^  +  e2e^.  Ans.  y"  -5y'  +  6y  =  0. 

1.   y  =  CiC^  +  e-z^'.  Ans.     y"  —  (a  +  h)y'  +  ahy  =  0. 

8.  xy  =^c  +  c-x.  Ans.     x*y'-  —  y'x  +  y. 

9.  y  =  {ci  +  x)e^'' +  c^e'.  Ans.     y"  -  ■iy'+3y=2  e^. 

10.  y  -  Cie^  +  Coe-^  +  c^e^'.  Ans.     y'"  -  Q  y"  +  \\y'  -Qy  =  0. 

11.  r  =  c  sin  d.  Ans.     rcosd  =  7-'  sin  6. 

12.  r=e'».  Ans.     r]ogr  =  r'0. 

13.  Assuming  the  differential  equation  found  in  Ex.  1,  indicate  the 
values  of  ?/'(=  —x/y)  at  a  large  number  of  points  (x,  y)  by  short  straight- 
line  segments  through  each  point  in  the  correct  direction.  Continue 
doing  this  at  points  distributed  over  the  plane  until  a  set  of  curves  is 
outlined.     Are  these  the  curves  given  in  Ex.  1  ? 

14.  Proceed  as  in  Ex.  13  for  the  equation  ?/'  =  y/x.  Do  you  recognize 
the  set  of  curves  ?    Can  you  prove  that  your  guess  is  correct  ? 

15.  Draw  a  figure  to  illustrate  the  meaning  of  y'  =  x^.  Find  y.  Gen- 
eralize the  problem  to  the  case  y'  =  /  (a;). 


352  DIFFERENTIAL  EQUATIONS  [X,  §  180 

16.  Find  that  curve  of  the  set  given  in  Ex.  1  which  passes  through 
(1,2).  Find  its  slope  (value  of  y')  at  that  point.  Do  these  three  values 
of  (x,  y,  y')  satisfy  the  differential  equation  given  as  the  ansvrer  in  No.  1  ? 

17.  Proceed  as  in  Ex.  16  for  the  equation  of  Ex.  2. 

18.  Proceed  as  in  Ex.  16  for  the  first  equation  of  Ex.  15. 

19.  Find  the  differential  equation  of  all  circles  having  their  centers  at 
the  origin. 

20.  Find  the  differential  equation  of  all  parabolas  with  given  latus 
rectum  and  axes  coincident  with  the  x-axis. 

21.  Find  the  differential  equation  of  all  parabolas  with  axes  falling  in 
the  X-axis. 

22.  Find  the  differential  equation  of  a  system  of  confocal  ellipses. 

23.  Find  the  differential  equation  of  a  system  of  confocal  hyperbolas. 

24.  Find  the  differential  equation  of  the  curves  in  which  the  sub- 
tangent  equals  the  abscissa  of  the  point  of  contact  of  the  tangent. 

25.  A  point  is  moving  at  each  instant  in  a  direction  whose  slope  equals 
the  abscissa  of  the  point.  Find  the  differential  equation  of  all  the  possi- 
ble paths. 

26.  Write  the  differential  equation  of  linear  motion  with  constant 
acceleration  ;  of  linear  motion  whose  acceleration  varies  as  the  square  of 
the  displacement.     The  same  for  angular  motion  of  rotation. 

27.  A  bullet  is  fired  from  a  gun.  Write  the  differential  equations 
which  govern  its  motion,  air  resistance  being  neglected.  How  must 
these  equations  be  modified,  if  air  resistance  is  assumed  proportional  to 
velocity  ? 

181.  General  Statement.  We  shall  now  consider  methods 
for  solving  differential  equations.  Since  the  most  common 
properties  of  curves  involve  slope  and  curvature,  and  since  in 
the  theory  of  motion  we  deal  constantly  with  speed  and 
acceleration,  the  differential  equations  of  the  first  and  second 
orders  are  of  prime  importance. 

Ordinary  differential  equations  of  the  first  order  and  first 
degree  have  the  form 

(1)  M-\-  N^  =  0,  or  Mdx  +  Ndy  =  0, 

dx 

where  M  and  N  are  functions  of  x  and  y. 


I 


X,  §  182j  SEPARATION  OF  VARIABLES  353 

No  general  method  is  known  for  solving  all  such  differential 
equations  in  terms  of  elementary  functions.  We  proceed  to 
give  some  standard  methods  of  solution  in  special  cases. 

182.    Type  I.     Separation  of  Variables.    It  may  happen  that 

Jf  involves  x  only,  and  N  involves  y  only.  The  variables  are 
then  said  to  be  separated  and  the  primitive  is  found  by  direct 
integration : 

^Mdx+  ^  N-dy  =  C, 

C  being  an  arbitrary  constant. 

Example  1.  Find  the  curves  having  a  constant  subnormal  equal  to  k. 
The  differential  equation  is 

subnormal  =  y  •  -^  =  k. 
dx 

Separating  the  variables  :  y  dy  =  k  dx. 

Integrating  both  sides  :       ^y'~  =  kx  +  c, 
or  2/2  =  2  kx  +  c', 

a  family  of  parabolas.  The  constant  c'  is  determined  if  the  parabola  is 
required  to  pass  through  some  given  point  in  the  plane. 

Check  this  result  by  eliminating  c  again  by  the  methods  of  §  179. 

Example  2.     Given  the  relative  rate  of  change  (logarithmic  derivative) 
of  a  function  of  x  in  terms  of  x,  find  the  function  :  i.e.  given 
(dy/dz)  -4-  2/  =  0(x),  to  find  y  =f(x). 
The  differential  equation 

^^y  =  <i>(x) 
dx 

is  of  the  type  mentioned  above  ;  separating  variables  and  then  integrating 

we  find  : 

-^  =  <f>(x)dx,  whence  log  y  =  ( <P  (x)  dx  +  c,  or  y  =  kJ*'-''"^, 

y  -^ 

where  k  =  e".  If  (t>(x)  =  x,  for  example,  y  =  ke^"/-;  if  also  the  value  of  y 
is  given  for  some  value  of  x,  say  y  =  .3,  when  x  =  2,  we  liave  3  =  ke^, 
whence  fc  =  3  e-2  and  w  =  3  e'-l'^-'^.     Check  this  result. 


354  DIFFERENTIAL  EQUATIONS  [X,  §  183 

183.  Type  II.  Homogeneous  Equations.  When  M  and  N 
are  homogeneous  *  in  x  and  y  and  of  the  same  degree,  the  equa- 
tion is  said  to  be  homogeneous.  If  we  write  the  equation  in  the 
form 

dy  _      M 
(£~      N' 
and  make  the  substitution 

dii  ,  xdv 

^  =  '"'''      dx^'^^'d^' 
we   obtain   a  new  equation   in   which   the   variables  can   be 
separated. 

Example  1. 
(1)  {xy  +  y2)  dx  +  {xy  -  x2)  dy  =  0, 


dy  _xy  +  y^ 
dx     x^  —  xy 


(2) 

Substituting  as  above  : 

(3)  ^^^(?l._«x2  +  .2^2_^  + 


dx      x''  —  vxP'       1  —  ■» 


^dv  ^  2v'^  . 
dx     1  —  t) ' 


separating  variables,  ^dv  —  — 

-Iv^  X 

Integrating  :  - log  v  =  log  a;  +  c. 


Keplacing  v  by  y/x, 


A_llog^  =  logx 
2u      2        X 


or  \ogxy  = 2c; 

y 

hence 

(4)  xy  =  e-'/'J-"^", 

or  xy  =  Ae-^/J', 
where  k  =  e-^^. 

*  Polynomials  are  homogeneous  in  a:  and  y  when  each  term  is  of  the  same 
degree.  In  general, /(x,  y)  is  homogeneous  if  f{kx,  ky)=  ^"/(a;,  y)  for  some 
one  value  of  n  and  for  all  values  of  k. 


I 


X,  §  183]  SEPARATION   OF  VARIABLES  355 

Check  :  Differentiating  both  sides  of  (4)  with  respect  to  a;,  we  find 

(5)  ydx  +  xdy^A-e-^/v[-y^y^^y]  ; 
dividing  the  two  sides  of  (5)  by  the  corresponding  sides  of  (4)  respectively 

(6)  [_ydx  +  xdy^^xy=-y^^^^=^', 
show  that  (6)  agrees  with  (1). 

EXERCISES  LXXm.  —  SEPARATION  OF  VARIABLES 

Solve  the  following  exercises  by  separating  the  variables : 

1.  xdy  +  y  dx  =  0.  Ans.   xy  =  c. 

2.  X Vl  +  y-  dx-y^\  +x'^dy  =  0.       Ans.    Vl  +  x^  =  Vl  +  y-  +  c. 

3.  sin  tf  (ir  +  r  cos  ^  (Z^  =  0.  Ans.    r&\ne  —  c. 

4.  xVl  +  y  dx  =  yVl  +  xdy. 

Solve  the  following  homogeneous  equations  : 

5.  (x  +  y)  dx+ (^x  —  y)  dy  =  0.  Ans.   x"^ -\- 2  xy  -  y~  —  c. 

6.  {x?  +  y')dx  =  2xydy.  Ans.   x^  -  y'^  =  ex. 

7.  (3  x2  —  y-)  dy  =  2xy  dx.  Ans.   x-  —  y"  =  cy^. 

8.  (x2  4-  2  xy  -  2/2)  ax  =  {x^-2xy-  2/2)  dy. 

Ans.   x2  +  y2  =  c(x+j/). 

The  following  Ex.  9-18  are  intended  partially  for  practice  in  recogniz- 
ing types : 


9.    Vl  —  y-dx  +  y/l  —  x2  dy  =  0.  Ans.   sin-'  x  +  sin-J  y  =  c. 

10.  x^  dx  +  (3  x2y  +  2  y3)  (^y  =  0.  Ans.   x^  +  2y'^  =  cy/di^ +yK 

11.  dy  +  y  sin  X  dx  =  sin  x  dx.  12.    rdd  =  tan  6  dr. 

13.    (2/  -  1)  dx  =  (x  +  1)  dy.  14.    ydx+  (x-y)dy  =  0. 

15.    X  (1  +  2/2)  dx  =  2/  (1  +  3^2)  d2/.  16.    («  ^2  +  2/^)  dx  =  2  xy  dy. 

17.   f^^x  =  c.  18.    ^^^y  =  x. 

dx  dx 

19.    In  Ex.  1  above,  draw  a  figure  to  represent  the  direction  of  the 

integral  curves  at  various  points.  Hence  solve  the  equation  geomet- 
rically. 


356  DIFFERENTIAL  EQUATIONS  [X,  §  183 

20.  A  point  moves  so  that  the  angle  between  the  x-axis  and  the  direc- 
tion of  the  motion  is  always  double  the  vectorial  angle.  Determine  the 
possible  paths.  ^^^         xy     ^  ^^    ^  >  0. 

X2  +  2/2 

21.  Proceed  as  in  Ex.  20  for  a  point  moving  so  that  its  radius  vector 
always  makes  equal  angles  with  the  direction  of  the  motion  and  the  x-axis. 

A71S.    r  =  c  sin  0. 

22.  The  speed  of  a  moving  point  varies  jointly  as  the  displacement 
and  the  sine  of  the  time.  Determine  the  displacement  in  terms  of  the 
time.  Ans.   s  =  ce-*<=08'. 

23.  Find  the  value  of  y  if  its  logarithmic  derivative  with  respect  to  x 
is  x2. 

184.  Type  III.  Linear  Equations.  This  name  is  applied  to 
equations  of  the  form 

(1)  |+i'.=  «, 

where  P  and  Q  do  not  involve  y,  but  may  contain  x.  Its  solu- 
tion can  be  obtained  by  first  finding  a  particular  solution  of 
the  reduced  equation, 

(1*)  ^  +  Py*==0, 

ax 

where  7j*  is  a  new  quantity  introduced  for  convenience  in  what 
follows ;  and  where  Q  is  replaced  by  zero.  In  (1*)  the  vari- 
ables can  be  separated  (see  Ex.  2,  §  182),  and  we  get 

jt  -\PdX 

y*  =e  •' 

as  a  particular  solution,  the  constant  C  of  integration  being 
given  the  particular  value  0. 

If  we  make  the  substitution 

(2)  y  =  V'y*j 


X,  §  184]  LINEAR  EQUATIONS  357 

« 
where  t>  is  a  function  of  x  to  be  determined,  the  equation  (1) 
becomes 

dx  dx 

The  first  term  vanishes  by  (1*)  leaving 

y*^  =  Q,   or   dv  =  ^dx==[Qj'""ldx. 
dx  y 

Hence 

v=     r-^c/x  +  c=   fcgJ^'^^jdaj  +  c 
and 

(3)  y=vy*  =  e-^'"'^^jlQe^'''yix  +  c 

This  equation  expresses  the  sohition  of  any  linear  equation. 
It  should  not  be  used  as  a  formula;  rather,  the  substitution 
(2)  should  be  made  in  each  example. 


Example  1.     Given 


dy 


(1)'  ^+3x22/  =  x6, 

dz 

the  reduced  equation  in  the  new  letter  y*  =  y/v  is 

(1*)'  ^  +  3  x^y*  =  0,     whence    y*  =  e"**. 

Hence  the  substitution  y  =  v  •  y*  becomes 

(2)'  y  =  ve-'\     whence     ^  =  e"*' ^"  -  3  va;2e-,»^ 

dx  dx 

and  (1)  takes  the  form 

Te-.-'l^  _  3  vx^e-^'l  +  3  ar^  [vg-x'']  =  arS. 

This  reduces,  as  we  foresaw  in  general  above,  to  the  form 

e-x»^  =  a:5     or    '^  =  xfie'', 
dx  dx 


358  DIFFERENTIAL  EQUATIONS  [X,  §  184 

whence 

V  -Ix^e^^dx  +  c  =  -I  [a;=e^=  -  e^']+  c, 

or,  returning  by  (2)  to  ?/ : 

(3) '  y  =  ve~^'  =  i  [a;3  -  1]  +  ce-< 

Check :    Differentiating  both  sides, 

(4)  ^  =  a;2-3x2ce-^*; 

dx 

eliminating  c  by  multiplying  (3)'  by  3  x^  and  adding  to  (4), 

^+3x2y=a;6. 
dx 

The  result  (3)'  may  also  be  obtained  by  direct  substitution  in  (3)  from 
(1)'.  Suificient  practice  in  the  direct  solution,  as  in  the  preceding  ex- 
ample, is  strongly  advised. 

185.  Extended  Linear  Equations.  This  name  is  often  given 
to  equations  of  the  form 

dy/dx+Py=  Qy\ 

Putting  z  =  ?/'""  reduces  it  to  a  linear  equation  in  z. 

Example  1.     Given 


^+y^xy^.     Tut  z  =  y-^. 
dx      X 

Then 

;^  =  -2,-3^,    or    f^  =  -(l/2),3^ 
dx                  dx  .         dx                       dx 

Thus 

-(l/2)2/3g  +  ^  =  x2/3 

and 

^_25=_2x. 
dx        X 

Here 

P  =  -?,    ^ Pdx  =-2\ogx,      eJ"^''"  =  x-2; 

so  that 

z  =  x^(  (  -  ^dx  +  cW-  2  x2  logx  +  cx2  =  2/-2, 

and  finally  x'^y-  (c  —  2  log  x)  =  1.     Check  this  result. 


X,  §186]  MISCELLANEOUS  EXERCISES  359 

EXERCISES    LXXIV.  — LINEAR   EQUATIONS 
Solve  the  following  linear  equations  and  check  each  answer : 

1.  ^Iy.—  xy  =  e'V2,  3,  ^  +  2/  cos  X  =  sin  2  x. 

dx  dx 

2.  ^  +  3  3-2^  =  3  3:5.  4.    .r  '!l>+y=  log  x. 
dx  dx 

Solve  the  following  extended  linear  equations,  checking  each  answer : 

5.  ^  +  y  =  ,f,  7.   ^  +  re  ^  r^  sin  e. 
dx     X  de 

6.  ^  +  M  =  .rw3.  8.    xy"-  '^-y^  =  x?. 
dx  dx 

Solve  the  following  equations,  checking  each  answer : 

9.   cos2  x^^  +  y  =  tan  x.  10.    /^^'  =  (l+r^)  sin  $. 

dx  dd 

11.    ^  =  -s  +  L  12.    ^+2/  =  .-. 

dt  dx 

Ans.    s  =  ce-'  —  1  +  «.  Ans.    ye^  =  x  +  c. 

13.   dy  -  ydx  =  !iinxdx.  14.    sec  »d/-  +  (r  -  1)^^  =  0. 

15.    (x^+l)dy=(xy  +  k)dx.  16.    a; («y  +  y rfx  =  .r»/2 log x dx. 

17.   The  equation  of  a  variable  electric  current  is 

L—+Bi=e, 
dt 

where  L  and  B  are  constants  of  the  circuit,  i  is  the  current,  and  e  the 
electromotive  force  of  the  circuit.  Calculate  i  in  terms  of  t,  1°,  if  e  is 
constant ;  2°,  if  e  =  eo  sin  tx)t. 

A71S.  2°     i  =  ^"  sm  (wt  -<f>)+  ce-i^/%  0  =  arc  tan  (w  L/i?) . 

186.  Other  Methods.  Nonlinear  Equations.  A  variety  of 
other  methods  are  given  in  treatises  ou  Differential  Equations; 
some  of  these  are  indicated  among  the  exercises  which  follow. 
Noteworthy  among  these  are  the  possibility  of  making  advan- 
tageous snbstitntions ;  and  —  what  amounts  to  a  special  type  of 
substitution  —  the  possibility  of  writing  the  given  equation  in 


360  DIFFERENTIAL  EQUATIONS  [X,  §  186 

the  form  of  a  total  differential,  dz  =  0,  where  2;  is  a  known 
function  of  x  and  y,  which  leads  to  the  general  solution 
z  =  constant  (see  Exs.  4-11,  below). 

Equations  not  linear  in  y'  may  often  be  solved.  If  the  given 
equation  can  be  solved  for  y',  several  values  of  y'  may  be  found, 
each  of  which  constitutes  a  differential  equation :  the  general 
solution  of  the  given  equation  means  the  totality  of  all  of  the 
solutions  of  all  of  these  new  equations  (see  Exs.  15-16,  p.  362). 

EXERCISES  LXXV.— MISCELLANEOUS  EXERCISES 

1.  Solve  the  equation  2  y  dy/dx  +  xy"^  =  e". 

[Hint.  Put  y'^  =  v;  then  dv/dx  =  '2,y  dy /dx,  and  the  equation  becomes 
dv/dx  +  xv^e",  which  can  be  solved  by  previous  methods.] 

2.  Solve  the  equation  cos  ydy  +  sin  y  sec^  x  dx  =  tan  x  dx. 

[Hint.  Put  v  =  siny,  ?/  =  tana;;  then  dv  =  cos  ydy,  du=»ec^xdx;  the 
equation  becomes  dv  +vdu  =  u  dn/(l  +  w^),  which  is  linear.] 

3.  Solve  the  following  equations,  using  the  indicated  substitutions: 
(a)  y^dy  +  (y^  +  x)  dx  =  0.     (Put  t^  =  y^  ) 

{b)  sdt  ~tds  =  2s{t- s)dt.     (Fut  s  =  tv.) 

(c)  xdy  —  ydx  =  {x--  y-)  dy.     (Put  y  =  vx.) 

(d)  m2«2  (udv  +  v  du)  =  (v  +  u2)  dv.    Put  uv  =  x,  V-  y.) 

4.  Solve  the  equation  (3  x^  +  y)  dx  +  (x  +  S  y^)  dy  =  0. 

[Hint.    If  we  put  z  =  x^  +  xi/  +  y«,  this  equation  reduces  to  dz  =  0;   for 
dz  =  idz/dx)dx  +  {dz/dy)dy.    But  dz=0  gives  z  =  const.,  hence  a:8  +  j;y  +  j,8=c    ^ 
is  the  general  solution.     Such  an  equation  as  that  given  in  this  example  is 
called  an  exact  differential  equation.] 

5.  Solve  the  equation  xdy  —  ydx  =  0. 

Hint.    This  equation  can  be  solved  by  previous  methods;  but  it  is  easier     | 
to  divide  both  sides  by  x^  and  notice  that  the  resulting  equation  is  d{y/x)  =  0;     { 
hence  the  general  solution  is  y/x  =  c.     A  factor  which  renders  an  equation 
exact  (l/a;2  in  this  example)  is  called  an  integrating  factor. 

6.  Solve  the  equation  (x^  +  2  xj/'^)  dx  +  (2  x-y  +  y-)  dy  =  0. 
[Hint.    Put  z  =  x^f-.i  +  x^y^  +  y^/S.] 


X,  §  186]  MISCELLANEOUS  EXERCISES  361 

7.  Solve  the  equation  (s  +  <  sin  s)  ds  +  (t  —  cos  s)  dt  =  0. 

[Hint.  Arrange:  sds+ [t  sin  sds  — cos  sdt]  +  tdt  =  0;  integrate  this 
knowing  that  the  bracketed  term  is  —  d{t cos s).] 

8.  Solve  the  equation  xdy  —  (y  —  x)  dx  =  0. 

[Hint.  Arrange:  [xdy  —  ydx^+xdx  =  0;  divide  by  x^,  and  compare 
Ex.  5.] 

9.  Show  that  [/(.r)  +  2  xy^]  dx  +  [2  x^y  +  <p  (y)]  dy  =  0  can  always  be 
solved  by  analogy  to  Ex.  6. 

10.  Show  that  [/(.'•)  +  y]  dx  —  xdy  can  always  be  solved  by  analogy 
io  Ex.  6.     Solve  (x-  +  y)  dx  —  x  dy  =  0.    Ans.  x  —  y/x  =  c. 

11.  Solve  the  equation  (r  —  tan  6)  dd  +  (r  sec  6  -\-  tan  d)dr  =  0. 
[Hint.     Multiply  both  sides  by  the  integrating  factor  cos^;   —sinedd 

+  rdr  +  d(r  sin  ^)  =  0  ;  integrate  term  by  term.] 

12.  When  a  family  of  curves  crosses  those  of  another  family  every- 
where at  right  angles,  the  curves  of  either  family  are  called  the  orthogonal 
trajectories  of  those  of  the  other  family. 

Find  the  orthogonal  trajectories  of  the  family  of  circles 

[Hint.  If  the  differential  equation  of  the  first  family  be  dy/dx  =f(x,  y), 
then  the  differential  equation  of  the  orthogonal  trajectories  is  dx/dy  =—/{x,  y) , 
for  at  any  point  of  intersection  (.r,  y)  the  slope  of  the  curve  of  one  system  is 
the  negative  reciprocal  of  the  slope  of  the  curve  of  the  other. 

In  this  example  the  differential  equation  of  the  given  family  is  x  dx+y  dy=0. 
It  is  evident  that  the  differential  equation  of  the  orthogonal  family  is  obtained 
by  replacing  dy  and  dx  by  —  dx  and  dy,  respectively ;  hence  the  desired 
equation  is  xdy  —  ydx  =  0,  whence  the  curves  are  y  =  ex,  i.e.  the  family  of 
all  straight  lines  through  the  origin.] 

13.  Find  the  orthogonal  trajectories  of  the  exponential  curves 

y  —  e^  +  k. 

[Hint.  The  differential  equation  is  dy/dx  =  e'.  The  orthogonal  family  is 
defined  by  the  equation  dy/dx  =  —  e-',  whence  the  trajectories  are  j/  =  e"'  +  c. 
Draw  the  figure.] 

14.  Determine  the  orthogonal  trajectories  of  the  following  families, 
and  draw  diagrams  in  illustration  of  each  : 

(a)  x-\-y  =  k  {d)  x2  +  t/^  =  2  log  a;  +  c. 

(b)  xy  =  k.  (e)  2x'^  +  y^  =  c^. 

(c)  1/2  =  4  ^ (a;  +  ^).  ^y ^  a.2  ^.  y2  =  jfx. 


362  DIFFERENTIAL   EQUATIONS  [X,  §  186 

15.  Solve  the  equation  y'~  —  (x  +  y)y'  +  xy  =  0,  where  y'  =  dy/dx. 
[Hint.    Solving  for  y'  we  find  y'  —  x  ov  y'  =  y.    The  general  solutions  of 

these  two  equations,  which  can  easily  he  found  by  previous  methods,  can  he 
written  together  in  one  equation  hy  a  principle  of  Analytic  Geometry: 
(22/-a;2-c)(2/-ce-)=O.J 

16.  Solve  each  of  the  following  equations : 

(a)  2/'2  _  4  y'a;  +  4x2  -  1  =  0.  Ans.  {y  -  x^y  -  (x  +  c)2  =  0. 

(6)  xY^  +  3  xyy'  +  2y'^  =  0.  Ans.  {xy  -  c)  {x-y  -  c)  =0. 

(c)  y'(y'  +  y)  =  x{x  +  y).   Ans.  (2  y  —  x'^  —  c)(y  +  x  -  1  -  ce-^)  =  0. 

{d)  j/2  +  yi2  -  1,  Ans.  y^  =  cos^  (x  +  c)/ 

(e)  y'-^-l  —  x2.  Ans.  2y  =  ±  (xVl  -  x'^  +  arc  sin  x)  +  c. 

17.  Solve  the  equation  2y  —  4  +  m^,  where  m  =  dy/dx. 

[Hint.  This  may  he  solved  as  above,  or  by  the  following  simpler  process : 
Differentiate  both  sides  with  respect  to  a; :  2  m  =  2  in  {dm/dx) ;  solve  this  for 
m:  m  =  x  +  c;  substitute  this  value  in  the  given  equation :  2  ?/  =  4  +  (x  +  c)2.] 

18.  Solve  the  equation  y  =  mx  +  nfi. 

[Hint.  Proceeding  as  in  Ex.  17,  we  find  that  the  new  equation  in  m  is: 
(x  +  2  m)  {dm/dx)  =  0.    If  dm/dx  =  0,  to  =  c ;  substituting :  y  =  cx+  c^.] 

19.  Find  the  envelope  of  the  solutions  of  (18)  and  show  that  the 
envelope  itself  is  a  solution. 

[Hint.  See  §  152,  p.  298.  The  envelope  is  y  =  -  xVi.  Show  this  is  a  solu- 
tion of  the  given  equation  (Ex.  18)  by  direct  check.  The  envelope  process 
demonstrates  this  also,  for  the  values  of  {x,  y,  m)  at  any  point  of  the  envelope 
are  the  same  as  {x,  y,  m)  on  some  curve  of  the  set  y=  cx-\-  c^.] 

20.  In  like  manner,  show  that  the  envelope  of  any  set  of  solutions  of 
any  differential  equation  is  itself  a  solution.  [This  new  solution  is  called 
a  singular  solution.] 

21.  Solve  the  following  equations,  find  the  envelope  of  the  solutions 
in  each  case,  and  show  that  the  envelope  is  a  solution.  [These  are  called 
Clairaut  equations.] 

2  X     

(a)  2/  =  mx  -  m3.  Ans.  y^cx-c^;  y  =  —VSx. 

(6)  y  =  mx-  e".  Ans.  y  =  ex  —  e<= ;  y  =  x\ogx  —  x. 

(c)  y  =  mx  +f{m). 

Ans.  y  =  ex  +/(c)  ;  Eliminate  c  with  x  +/'  (c)  =  0. 


X,  §  188]  SECOND  ORDER  363 


PART   II.     ORDINARY   DIFFERENTIAL   EQUATIONS 
OF   THE   SECOND   ORDER 

187.  Special  Types.  We  first  consider  some  very  special 
forms  of  equations  of  the  second  order  that  are  most  frequently 
used  in  the  application  of  mathematics  to  physics,  namely : 

[I]  d^i^^  ^"'^    1^^'  =  constant]. 

[II]  ^S  +  -^|f+^^  =  ^   [A -B,  C,  constants]. 

[III]  ^^  +  ^11+^^  =  ^^^'^-     [^^  ^>  C-^  constants.] 

These  are  all  special  forms  of  the  general  equation  of  the  sec- 
ond order  <^{x,  y,  dy/dx,  d-y/dx^)  =  0. 

[IV]  We  shall  consider  other  special  forms  also,  some  of 
which  include  the  above;  namely,  the  cases  that  arise  when 
one  or  more  of  the  quantities  x,  y,  dy/dx,  are  absent  from  the 
equation.     (See  §  191,  p.  371.) 

188.  Type  I.  This  type  of  equation  arises  in  problems  on 
motion  in  which  the  tangential  acceleration  d?s/df  is  propor- 
tional to  the  distance  passed  over  (see  §  89,  p.  156)  : 

(1)  g=±*-, 

a  form  which  is  equivalent  to  [I],  written  in  the  letters  s  and 
t.  If  we  multiply  both  sides  of  tins  equation  by  the  speed 
V  =  ds/dt  and  then  integrate  with  respect  to  t,  we  obtain 

(2)  I -dt=  I  ±k-s—dt: 

^  ^  J  dt  de         J  dt      ' 

but  we  know  that 

J  dtde         J     dt         J  2  2\dt) 


364  DIFFERENTIAL  EQUATIONS  [X,  §  188 

and 

r±  kh  ~dt  =  ±  k^  Csds=  ±  Ms2  +  c' ; 

hence  (2)  becomes* 
Case  1.   If  the  sign  before  /cs  is  +,  (3)  becomes 


(4)  v='^^  =  kVs'+C„ 

whence  f      ^^       =  Ckdt  +  Co, 


(5)  log(s4-Vs2+Ci)  =  A:«+C2; 
or,  solving  for  s, 

(6)  s  =  Ae^*  4-  Be-'^', 

where   2  A  =  e^''   and   2  B  =  —  Cie'^'  are   two   new  arbitrary- 
constants. 

By  means  of  the  hyperbolic  functions  sinh  ?i  =  (e"  —  e-")/2 
and  cosh  ■«  =  (e"  +  e~")/2  this  result  may  also  be  written  in 
the  form 

(7)  s  =  a  sinh  (kt)  +  b  cosh  (kt), 
where                      b  +  a=  2  A  and  h  —  a  —  2  B. 

Case  2,  If  the  sign  before  k^  is  — ,  Ci  must  be  negative 
also,  or  else  v  is  imaginary ;  hence  we  set  Cj  =  —  a^  and  write 

(42)  v=   i^  =  ^V(?^^ 


kdt  +  O2, 


*  This  is  often  called  the  energy  integral,  for  if  we  multiply  through  hy 
the  mass  m,  the  expression  mv^/2  on  the  left  is  precisely  the  kinetic  energy  of 
the  body. 


X,  §  188]  SECOND  ORDER  365 

whence 

or  solving  for  s : 

(62)  s  =  a  sin  ( kt  +  C-i)  =  A  sin  kt -\-  B  cos  kt, 

where  A=:  a  cos  Cj  and  B  =  a  sin  C2  are  two  new  arbitrary 
constants. 

Equation  (6,)  is  the  characteristic  equation  of  simple  har- 
monic motion ;  the  amplitude  of  the  motion  is  a,  the  period  is 
2  ir/k,  and  the  phase  is  —  Cj/k. 

The  differential  equation  (1)  was  first  found  in  §  88,  p.  155, 
"We  now  see  that  the  general  simple  harmonic  motion  (60)  is 
the  only  possible  motion  in  which  the  tangential  acceleration 
is  a  negative  constant  times  the  distance  from  a  fixed  point ; 
i.e.  it  is  the  only  possible  type  of  natural  vibration  under  the 
assumptions  of  §  90,  p.  157. 

EXERCISES   LXXVI.  — TYPE  I 

1.  Solve  each  of  the  following  equations : 

(c)g=4..  (<:=-»- 

2.  Find  the  curves  for  which  the  flexion  (cPy/dx-)  is  proportional  to 
the  height  (y). 

3.  Determine  the  motion  described  by  the  equation  of  Ex,  1  (a)  if  the 
speed  V  (  =  ds/iU)  and  the  distance  traversed  s  are  both  zero  when  t  =  0. 

4.  Proceed  as  in  Ex.  3  for  Ex.  1  (b),  and  explain  your  result. 

5.  Write  the  solution  of  Ex.  1  (a)  in  terms  of  sinh  t  and  cosh  t.  De- 
termine the  arbitrary  constants  by  the  conditions  of  Ex.  3,  and  show  that 
the  final  answer  agrees  precisely  with  that  of  Ex.  3. 

6.  Determine  the  motion  described  by  the  equation  of  Ex.  1  (b)  if 
V  =  2  and  s  =  10  when  «  =  0  :  if  t?  =  0  and  s  =  5  when  t  =  0. 


366  DIFFERENTIAL   EQUATIONS  [X,  §  189 

189.  Type  II.  Homogeneous  Linear  Equations  of  the  Second 
Order  with  Constant  Coefficients.     The  form  of  this  equation  is 

where  A,  B,  C  are  constants. 

The  type  just  considered  is  a  special  case  of  this.  one.  Fol- 
lowing the  indications  of  the  results  we  obtained  in  §  188,  it 
is  natural  to  ask  whether  there  are  solutions  of  any  one  of  the 
types  we  found  in  the  special  case : 

Trial  of  e^^.  If  we  substitute  2/  =  e**  in  (1)  we  obtain  the 
equation : 

(2)  \_Ak-  +  Bk+(J]e^  =  0. 

The  factor  e**  is  never  zero ;  hence  k  must  satisfy  the  quad- 
ratic equation 
(1*)  Ak'2  +  Bk  +  C=0, 

which  is  called  the  auxiliary  equation  to  (1).  If  the  roots  of 
(1  *)  are  real  and  distinct,  i.e.  if 

(3)  D  =  B'-4.AC>0, 

then  these  roots  ki  and  k^  are  possible  values  for  k,  and  the  gen- 
eral solution  of  (1)  is 

(4)  y  =  Cie^'i'*'  +  Cae'^^"', 

since  a  trial  is  sufficient  to  convince  one  that  the  sum  of  two 
solutions  of  (1)  is  also  a  solution  of  (1) ;  and  that  a  constant 
times  a  solution  is  also  a  solution. 

Trial  oi  y  =  e^^  •  v.     If  (3)  is  not  satisfied,  the  substitution 

(5)  y=e'''  -v 
changes  (1)  to  the  form 

(6)  ^g  +  [2K.4  +  ^]^  +  [.lK^+5K+C].  =  0, 


X,  §  189]  SECOND  ORDER  367 

which  becomes  quite  simple  if  we  determine  k  so  that   the 
term  in  clv/dx  is  zero  : 

(7)  2kA+B=0,     whence     k  =  -B/2A; 

then  (6)  takes  the  form 

^  ^  d-r^  4.4-  ' 


where  K  =  V4  AC  -  B'/(2  A)=^-D/2A  is  real  if 

(9)  0=^^-4.  AC<0, 
zohich  is  the  case  ive  could  not  solve  before. 

If  D<0,  the  solutions  of  (8)  are 

(10)  v=Ci  sin  (K.r)  +  C.2  cos  (Ka;), 

by  (62),  §  188,  p.  365 ;  hence  the  solutions  of  (1)  are 

(11)  y  =  e>^v~e*i^[Ci  sin  (Kx)  +  Co  cos  (Ka?)], 

where  k  =  -  B/{2  A)  and  K  =  V^D/(2  A) ;  these  values  of  k 
and  K  are  most  readily  found  by  solving  (1  *)  for  A;,  since  the 
solutions  of(l*)arek  =  (-B±  VI>)/(2  ^)  =  k  ±  K  V^l. 

If  i>  =  0,  K  =  0  and  the  solutions  of  (8)  are 

(12)  v=CiX+C2; 
hence  the  solutions  of  (1)  are 

(13)  J/  =  eKx  .  V  =  cKa;[CiX  +  C'2], 

where  k  =  —B/(2  A)  is  the  solution  of  (1  *)  ;  since  when  D  =  0, 
(1  *)  has  only  one  root  k  =  ~  B/{2  A). 

It  follows  that  the  solutions  of  (1)  are  surely  of  one  of  the  three 
forms  (4),  (11),  (13),  according  as  D  =  R- -i  AC  is  +,  ~,  or 
0 ;  that  is,  according  as  the  roots  of  the  auxiliary  equation  (1  *) 
are  real  and  distinct,  imaginary,  or  equal;  in  resume: 


368 


DIFFERENTIAL  EQUATIONS 


[X,  §  189 


D=B^-iAC 

Character  of 
KOOTS  OF   (1») 

Values  of  Roots 
OF  (1*) 

Solution  of  (1) 

+ 

Real,  unequal 

fci,  k.2 

(4) 

- 

Imaginary 

K  ±  K  V^ 

(11) 

0 

Equal 

K 

(13) 

Such  solutions  as  (11)  have  been  forecasted  from  the  work 
of  §  92,  p.  162,  where  an  equation  (in  the  letters  s  and  t)  of  pre- 
cisely the  type  (11)  was  studied.  Indeed,  if  y  and  x  are  re- 
placed by  s  and  t,  and  if  k  is  negative,  (11)  expresses  precisely 
the  most  general  form  of  damped  vibration,  studied  in  §  92. 


Examples 

1 

2 

3 

Equation  (1) 

3j/"_4  2/'+2/=0 

32/"-4y+|2/=0 

3  2/"-42/'-H2  2/=0 

Auxiliary  equa- 
tion (1*) 

3A;2-4A:+1=0 

3*2-4  A:+f=0 

3i-2-4A-  +  2=0 

Roots  of  (1*) 

1,  1/3 

2/3,  2/3 

K2±V^) 

Solution  of  (1) 

2/=cie^+C2e^/3 

2/  =  e2x/3^Ci  +  C2X) 

?/  =  e2'/3(CiC0S 

EXERCISES   LXXVII.  —  LINEAR  HOMOGENEOUS.     TYPE   H 


1.  2/"  -  4  J/'  +  3  2/  =  0. 

2.  2/"  +  32/'  +  22/  =  0. 

3.  5  y"  -  4  2/' -}- 2/ =  0. 

4.  9  ?/"  +  12  2/'  +  4  2/  =  0. 

5.  y"  -2y>  +  y  =  0. 

6.  y"  +  y'  +  y  =  o. 

7.  y"  -2y'  +  3y  =  0. 

8.  3  J/"  +  6  2/'  -h  2  2/  =  0. 


9.  2/''  -  9  2/'  +  14  y  =  0. 

10.  2  2/"  —  3  y'  +  2/  =  0. 

11.  6  2/"  -  13  2/' +  6  2/ =  0. 

12.  2/"-3  2/'  =  0. 

13.  2/"-4  2/  =  0. 

14.  y"  +  9y  =  0. 

15.  2/"  +  A-2/'  =  0. 

16.  ?/"  ±  *y  =  0. 


X,  §  190]  SECOND  ORDER  369 

17.  If  a  particle  is  acted  on  by  a  force  that  varies  as  the  distance  and 
by  a  resistance  proportional  to  its  speed,  the  differential  equation  of  its 
motion  is 

d-r/dt-  +  h  dx/dt  +  ex  =  0, 

where  c  >  0  if  the  force  attracts,  and  c  <  0  if  the  force  repels.     Solve  the 
equation  in  each  case. 

18.  If  in  Ex.  17,  6  =  c  =  1,  and  the  particle  starts  from  rest  at  a  dis- 
tance 1,  determine  its  distance  and  speed  at  any  time  t.  Is  the  motion 
oscillatory  ?  If  so,  what  is  the  period  ?  Solve  when  the  initial  speed 
is  Vp. 

19.  If  in  Ex.  17,  6  =  1  and  c  =—  1,  discuss  the  motion  as  in  Ex.  18. 


190.   Type  III.    Non-homogeneous  Equations.     This  type  is 
of  the  form : 

where  A,  B,  C,  are  constants,  and  F(x)  is  a  function  of  x  only. 
We  proceed  to  show  that  this  form  can  be  solved  in  a  manner 
exactly  analogous  to  §  184,  p.  356 ;  first  write  down  the  reduced 
equation  in  the  new  letter  y* : 

doi?  dx 

and  solve  (1*)  by  the  method  of  §  189.  Let  y*  =  <^(x)  be  any 
one  particular  solution  of  (1*)  (the  simpler,  the  better,  except 
that  ?/*  =  0  is  excluded).     Then  the  substitution 

(2)  y  =  <i>{x).u 
transforms  (1)  into 

(3)  \A<i>"{:x)  +  B<^\x)  +  C4>{x)\u+  \2A<i>\x)+  B<i>{x)\^ 

+  A^(x)p{=F{x); 
ax^ 

2b 


370  DIFFERENTIAL   EQUATIONS  [X,  §  190 

but,  since  <f>{x)  satisfies  (1*),  the  first  term  of  (3)  is  zero ;  and 
if  we  now  set  du/dx  =  v  temporarily,  this  equation  can  be 
written  as  the  linear  equation : 

.^.  dv       (2A^'(x)  +  B<f>(x^  ^.^  F(x) 

^  ^  dx      \  A<l>(x)  j  A4>(xy 

which  is  precisely  of  the  form  solved  in  §  184.  Comparing  (4) 
with  (1),  §  184,  we  have 

Having  found  -y  by  §  184,  we  have 

u=  i  vdx  +c^,      y  =  u<t>{x)  =  <f> (ic)    j  v <?a?  +  Cg  , 
which  is  the  required  solution  of  (1). 
Example  1.     Given  the  equation 

we  write  the  reduced  equation 

(1*)  f^+3lj?+2y*  =  0; 

dx^  ax 

this  is  easily  solved  by  the  method  of  §  189  ;  the  simplest  particular  solu- 
tion IS,  y  =  e-'.  Substituting  0(a;)  =  e~'  in  the  general  work  above,  we 
find 

p^2^0MJl^K^  =  land§=^^:M.  =  e«sina:; 
^0(x)  ^      A<p(:x) 

hence  e-''^'*'  —  e',  and 

V  =  e-^l   \  e^x  sin  xdx-\-  Cx\  =  -  c^(2  sin  x  —  cos  a;)  +  Cie-», 

u=   \  V  (?x  +  Co  =  —  e' (sin  x  —  3  cos  x)  —  CiC~^  +  Coe 
y  =  u-  <p(x)  —  — (sin  x  —  3  cosx)  —  Cie~^'  +  C^-'. 


X,  §  191J  SECOND   ORDER  371 

EXERCISES  LXX VIII.  — NON-HOMOGENEOUS   TYPE 

1.  y"  —  3  y'  +  2  y  =  cos  x. 

Ans.   y  =  ^  (cos  x  —  3  sin  a;)  +  ciC  +  c^e^. 

2.  y"  -iy'  +  2y  ^  X. 

Ans.  y  =  ^(x  +  2)  +  Cie(2+''2)^  +  Cze^*-^'^)'. 

3.  y"  +  Sy>  +2y  =  e'. 

Ans.   y  =  eV6  —  Cie-^*  +  c^e-'. 

4.  y"  -2y'  +y=  z.  Ans.   y  =  x+2  +  e=^(ci  +  Czx). 

5.  y"  +  y  =  sinx.  Ans.    ?/ =— |a;cos  x  +  Cisinx  +  C2C0SX. 

6.  y"  —  y'  —  2y  =  sin x. 

Ans.   y  —^^{cos,x  —  3sinx)  +  Cig-^  +  o^e^'. 

7.  2/"  +  4  y  =  x2  +  cos  X. 

Ans.   2/  =  ^(2  x2  —  1)  +  1  cosx  +  Ci  cos  2  x  +  C2  sin  2  x. 

8.  y"  -2y'  =  e^'  +  1.  ^ms.   y  =  ^  x(e2*  _  1)  +  d  +  C2e2». 

9.  J/"  -  4  y'  +  3  2/  =  2  e^^.  .4«s.   y  =  xe*=  +  Cic'  +  de^. 
10.   If  a  particle  moves  under  tlie  action  of  a  periodic  force  through  a 

medium  resisting  as  the  speed,  the  equation  of  motion  is 

d-s/dfi  +  Ads/dt  =  5  sin  C  t. 

Express  s  and  the  speed  in  terms  of  «.  If  ^  =:  5  =  C  =  1,  what  is  the 
distance  passed  over  and  the  speed  after  5  seconds,  the  particle  starting 
from  rest  ? 

191.   Type  IV.    One  of  the  quantities  x,  y,  y'  absent. 

Type  IV„ :  <t>{y")  =  0.  Solve  for  y",  to  obtaiu  a  solution,  say 
y"  =  a.  Then  integrate  twice.  The  general  solution  for 
each  value  of  y"  is  of  the  form  y  =  ^  axr  +  CiX  +  C2. 

In  problems  of  motion,  this  type  is  equivalent  to  the  statement 
that  <p(j^)  =  0,  whereby  =  d^s/dl"^  =  dv/d't.  Hence  j^, may  have  any  one 
of  the  several  constant  values' wliich  satisfy  <p(jj)=0;  but  if  jj,=  k, 
s  =  kt-/2  +  at  +  C2  (see  Ex.  4,  p.  73). 

Type  rVft :  y  missing.  <t>(x,  y',  y")  =  0.  The  substitution 
m  =  y'  z=dy/(lx,  fbn/d.v  =  d-y/dxr  =  y",  reduces  the  given  equa- 
tion to  an  equation  of  the^rs^  order  in  ?».  x,  dm/dx.  Solving, 
if  possible,  one  gets  a  relation  of  the  form  /(»i,  x,  c)  =  0.     This 


372  DIFFERENTIAL  EQUATIONS  [X,  §  191 

is  again  an  equation  of  the  first  order  in  x  and  y,  and  may  be 
integrated  by  methods  given  in  Part  I,  §§  182-186. 

The  interpretation  in  motion  problems  is  particularly  vivid  and  beauti- 
ful. Thus  V  =  ds/dt  and  j^  =  dv/dt  =  d?s/dt'^ ;  hence  any  equation  in 
j„,  V,  t,  with  s  absent,  is  a  differential  equation  of  the  first  order  in  v. 
Solving  this,  we  get  an  equation  in  v  and  t ;  since  v  —  ds/dt,  this  new 
equation  is  of  the  first  order  in  s  and  t. 

Example  1.  1  +  x  +  x2  ^^  =  0. 

dx^ 

Setting  dy/dx  =  m,      1  +  x  +  x'^  ^  =  0. 
dx 

1  4-  X 
Separating  variables,  —  dtn  =    ^    dx. 

x2 

Integrating,  —  m  = 1-  log  x  +  Ci. 

X 

Integrating  again,  y  =  log  x  —  x  log  x  +  (1  —  Ci)x  —  C2. 

Interpret  this  as  a  problem  in  motion,  with  s  and  t  in  place  of  y  and  x, 

and  jr  =  dv/dt  =  d^s/dt"^. 

Example  2.  In  a  certain  motion  the  space  passed  over  s,  the  speed  v, 
and  the  acceleration  j^,  are  connected  with  the  time  by  the  relation 
1  +  -y^  —  jy  =  0  ;  find  s  in  terms  of  t. 

Placing  j   =  dv/dt,  the  equation 


dt 
is  of  the  first  order.     The  variables  can  be  separated,  and  the  integral  is 

tan-i  V  =  t  +  cior  V  =  tan  (t  +  Ci), 
which  is  itself  a  differential  equation  of  the  first  order  if  we  replace  v  by 
ds/dt.     Integrating  this  new  equation  : 

(ds  =  ftan  (t  +  ci)dt  +  c-z,  or  s  =  -  log  cos  {t  +  c{)  +  c^. 

In  such  a  motion  problem  we  usually  know  the  values  of  v  and  s  for 
some  value  oi  t.  If  v  =  0  and  .s  =10  when  «  =  0,  for  example,  Ci  must  be 
zero  (or  else  a  multiple  of  tt)  and  C2  must  be  10  ;  hence  s=  —  log  cos  t-\r  10. 

Example^.  1  +  x  ^  +  x2^=  0  =1  +  xw  +  x^^. 

dx         dx^  dx 


This  can  be  written         dm/dx  ■\-m/x  —  —  l/x^ 


X,  §  191]  SECOND  ORDER  373 

which  is  linear  in  m  and  x,  the  solution  being 
m  =  —  log  X  +  -^-i . 

X  X 

The  second  integration  gives 

y=-i  [log  X]2  +  Ci  log  X  +  C2. 

Interpret  this  as  a  motion  problem,  and  determine  Ci  and  co  to  make 
y  =  10  and  m  =  3  when  x  =  1. 

T3rpe  IVc :  jc  missing.     <f>(y,  y',  y")  =  0.     The  substitution 
m  =  y'  gives 

,  ,,      fZu'      dy'     dy      dm 

dx       dy     dx      dy 

and  the  transformed  equation  is  an  equation  of  the  first  order 
in  y  and  m.  We  solve  this  and  then  restore  y'  in  place  of  m, 
whereupon  we  have  left  to  solve  another  equation  (in  x  and  y) 
of  the  first  order. 

This  is  precisely  the  way  in  which  we  solved  Type  I,  §  188, 
Type  I  being  only  an  important  special  case  of  Type  IV^. 

Example  1.     If  the  acceleration  jr  is  given  in  terms  of  the  distance 
passed  over  (compare  §  188),  we  have 

This  is  transformed  by  the  relation 

dv     dv  ds     dv 


(which  is  itself  a  most  valuable  formula)  into 

in  which  the  variables  can  be  separated  ;  integration  gives 

}v'^=(<t>{s)ds  +  c, 
which  is  called  the  energy  integral  (see  footnote,  p.  364). 


374  DIFFERENTIAL  EQUATIONS  [X,  §  191 

The  work  cannot  be  carried  further  than  this  without  knowing  an 
exact  expression  for  (/>(s).  When  0(s)  is  given,  we  proceed  as  in  §  188, 
replacing  v  by  ds/dt  and  integrating  the  new  equation  : 


J 


^^■2^,p(s)ds 


=  t+k. 


+  2c 


Unfortunately  the  indicated  integrations  are  difficult  in  many  cases ; 
often  they  can  be  performed  by  means  of  a  table  of  integrals.  One  case 
in  which  the  integrations  are  comparatively  easy  is  that  already  done  in 
§  188. 

EXERCISES  LXXIX.  — TYPE  IV 

1.  w"2- 4x2  =  0. 


Ans 

•  y  = 

±  l/3x3  +  CiX+C2. 

Ans 

.2y 

=  cie-  +  e-Vci  +  Co. 

Ans 

■  y  = 

XV9  +  Ci  log  X  +  C2. 

Ans 

.    S2  = 

:  «2  +  Cit  +  C2. 

6. 

da;2 

=  ±  k^y. 

8. 

dx^~ 

^e^y. 

10. 

dx^ 

=  X  +  3  sin  x. 

12. 

doi^' 

=['-(l)T- 

2.  2/"  =  Vl  +  2/'2. 

3.  xy"  +  2/'  =  x2. 

4.  s&^sldf^^idsldty^^  1. 

5.  ^  =  1-. 

dv^     Vi 

'    dx2-'    • 

9.  ^  =  x2cosx. 
dx2 

11.  ^  =  e^-cos2x. 
dx* 

13.  Show  that  Ex.  12  is  equivalent  to  the  problem,  to  find  a  curve 
whose  radius  of  curvature  is  unity. 

14.  The  flexion  {d^y/dx^)  of  a  beam  rigidly  embedded  at  one  end,  and 
loaded  at  the  other  end,  which  is  unsupported,  is  k{l  —  x),  where  A;  is  a 
constant  and  I  is  the  length  of  the  beam.  Find  y,  and  determine  the 
constants  of  integration  from  the  fact  that  ?/  =  0  and  dy/dx  =  0  at  the 
embedded  end,  where  x  =  0. 

15.  Find  the  form  of  a  uniformly  loaded  beam  of  length  Z,  embedded 
at  one  end  only,  if  the  flexion  is  proportional  to  ^2  _  2  Zx  +  x2,  where 
X  =  0  at  the  embedded  end. 

16.  Find  the  form  of  a  uniformly  loaded  beam  of  length  I,  freely  sup- 
ported at  both  ends,  if  the  flexion  is  proportional  to  P  —  4  x2  in  each  half, 
where  x  is  measured  horizontally  from  the  center  of  the  beam. 


X,  §  193]  HIGHER  ORDER  375 


PART   III.     GENERALIZATIONS 

192.  Ordinary  Equations  of  Higher  Order.  An  equation 
whose  order  is  greater  than  two  is  called  an  equation  of  higher 
order;  the  reason  for  this  is  the  comparative  rarity  in  applica- 
tions of  equations  above  the  second  order.  There  seems  to  be 
a  natural  line  of  division  between  order  two  and  higher  orders, 
which  is  analogous  to  the  natural  demarkation  between  space 
of  three  dimensions  and  space  of  higher  dimensions. 

We  shall  state  briefly  the  generalizations  to  equations  of 
higher  order,  however,  since  they  do  occur  in  a  few  problems, 
and  since  it  is  interesting  to  know  that  practically  the  same  rules 
apply  in  certain  types  for  higher  orders  as  those  we  found  for 
order  two. 

193.  Linear  Homogeneous  Type.  The  work  of  §  189  can  be 
generalized  to  any  linear  homogeneous  equation  with  constant 
coeflBicients : 

Thus  if  we  set  y  =  e*',  as  in  §  189,  we  find 

(1*)  A;"  +  ajfc"-'  +  •  •  •  +  a„_,  k  +  a„  =  0, 

again  called  the  auxiliary  equation.  Corresponding  to  any 
real  root  k^  there  is  therefore  a  solution  e''i';  if  all  the  roots  are 
real  and  distinct,  the  general  solution  of  (1)  is 

(2)  7/=(7,e*i^  +  C,e*^  +  ..-  +  C,.eV, 

where  k^,  k,,  •••,  k\  are  the  roots  of  (1).  Curiously  enough,  the 
chief  difficulty  is  not  in  any  operation  of  the  Calculus ;  rather 
it  is  in  solving  the  algebraic  equation  (1*). 

It  is  easy  to  show  by  extensions  of  the  methods  of  §  189 


376  DIFFERENTIAL  EQUATIONS  [X,  §  193 

that  any  pair  of  imaginary  roots  of  (1*),  A;=k±KV— 1  cor- 
responds to  a  solution  of  the  form  f 

(3)  2/  =  e"  [  C  sin  (A"^)  +  C '  cos  (Kx)], 

which  then  takes  the  place  of  two  of  the  terms  of  (2). 

Finally,  if  a  root  k  =  k  of  (1*)  occurs  more  than  once,  i.e.  if 
the  left-hand  side  of  (1*)  has  a  factor  (k  —  k)^,  the  correspond- 
ing solution  obtained  as  above  shoidd  be  multiplied  by  the 
polynomial 

(4)  Bo-hB,x-hB2x'-\--'-{-  B^_,xP-\ 

where  p  is  the  order  of  multiplicity  of  the  root  (i.e.  the  expo- 
nent of  (k  —  k)"),  and  where  the  B's  are  arbitrary  constants 
which  replace  those  lost  from  (2)  by  the  condensation  of  several 
terms  into  one. 

The  proof  is  most  easily  effected  by  making  tlie  substitution  y  =  e'^'  •  m, 
whereupon  the  transformed  differential  equation  contains  no  derivative 
below  d^u/dx^  ;  hence  u  =  the  polynomial  (4)  is  a  solution  of  the  new 
equation,  and  y  =  e**  times  the  polynomial  (4)  is  a  solution  of  (1).  This 
work  may  be  carried  out  by  the  student  in  any  example  below  in  which 
(1*)  has  multiple  roots.  J: 

t  This  fact  is  often  made  plausible  by  the  use  of  the  equations 
qv.\/—i  —  COSM+ V— Isinw,  e~"N/-i=  cos  u  —  V—1  sin  u ; 
these  equations  can  be  derived  formally  by  using  the  Taylor  series  for  e",  cos  u, 
sin  u,  vrith  v=uV—i,  but  they  remain  only  plausible  until  ^fter  a  study  of 
the  theory  of  imaginary  numbers.    The  solutions  e*  ±  A'V— i    are  indicated 
formally  by  (2) ;  hence  it  is  plausible  that  (3)  is  correct. 

A  more  direct  process  which  avoids  any  uncertainty  concerning  imaginaries 
is  almost  as  easy.  For  the  substitution  ?/  =  e«'«  (see  §  189)  gives  a  new 
equation  in  u  and  x  which,  together  with  its  auxiliary,  has  coefficients  of  the 
form  {d'>^A(k)/dk")  -f-n!,  where  A{k)  represents  the  left-hand  side  of  (1*). 
Now  5"  V— 1  is  a  solution  of  the  new  auxiliary  by  development  of  A{k)  in 
powers  of  (k  —  K) ;  hence  u  =  sin  (Kx)  and  (t  =  cos  (Kx)  are  solutions  of  the 
new  differential  equation,  as  a  comparison  of  coefficients  demonstrates.  This 
process  constitutes  a  rigorous  proof  of  (3) . 

J  To  avoid  using  imaginary  powers  of  e,  if  that  is  desired,  substitute 
y  =  e''''  [cos  (Kx)  -»-  V—  1  sin  (K^)]u,  when  the  multiple  root  is  imaginary, 


X,  §  194] 


HIGHER  ORDER 


377 


These  extensions  of  §  189  should  be  verified  by  the  student  by  a  direct 
check  in  each  exercise. 


KXAMPI.K 

1 

2 

3 

(1) 

2/"'-2/'=0 

yiv+6j/"'+12y"+8  2/'=0 

2/"'  +  8r/  =  0 

(1*) 

^•3  -  ^•  =  0 

A:*+6F+12A:2  +  8A;  =  0 

A,-3  +  8  =  0 

k  = 

0,  1,  -  1 

0,-2,-2,-2 

-2,  1±V3\/^1 

y 

ci  +  coc^  +  cse-* 

ci  4-  e-'^Ccs  +  C33;  +  dx'^) 

Cie-^  +  c^(c2Cos\/3x 

+  C3  sin  V3  x) 

194.   Non-homogeneous  Tjrpe.     The  non-homogeneous  type 


(1) 


T^  +  "1  TVl  +  •  • '  +  ""-1  7    +  ^^  = -^(^) 
dx"  dx"  ^  dx 


cannot  be  solved  in  general  by  an  extension  of  §  190.  But  in 
the  majority  of  cases  which  actually  arise  in  practice,*  a  suffi- 
cient method  consists  in  differ entiatiyg  both  sides  of  (1)  re- 
peatedly until  an  elimination  of  the  ?v(//<^hand  sides  becomes 
possible.     The  new  equation  will  be  of  higher  order  still : 


(2) 


dx""  dx""'^ 


^A 


dy 


+  -L  =  0, 


hd  its  rigid-hand  side  is  zero.  Solve  this  equation  by  §  193, 
and  then  substitute  the-  result  in  (1)  for  trial ;  of  course  there 
5^~will  be  too  many  arbitrary  constants;  the  superfluous  ones  are 
determined  by  comparison  of  coefficients,  as  in  the  examples 
below. 

Example  1.     y'"  +  y'  =  sinx. 

Differentiating  both  sides  twice   and   adding  the  result  to  the  given 

equation  : 

yy  +  2y"'  +  y'  =  0. 

*  For  more  general  methods,  see  any  work  on  Differential  Equations  ;  e.g. 
Forsyth,  Differential  Equations. 


378  DIFFERENTIAL  EQUATIONS  [X,  §  194 

The  auxiliary  equation  k^  +  2k^-{-k  =  0  has  the  roots  k=0,  k  =  ±  V'^ 
(twice).  Hence  we  first  write  as  a  trial  solution  y,  the  solution  of  the 
new  equation  :  y^  =  ci  +  (C2  +  C3X)  cos  x  +  (C4  +  Cscc)  sin  x  ;  substituting 
this  in  the  given  equation,  we  find  —  2  C3  cos  x  —  2  C5  sin  x  =  sin  x,  whence 
C3  =  0  and  C5  =  —  1/2 ;  substituting  these  values  in  the  trial  solution  y, 
gives  the  general  solution  of  the  given  equation  : 

</  =  Ci  +  Co  cos  X  +  (C4  —  x/2)  sin  x. 

EXERCISES  LXXX.— LINEAR  EQUATIONS  OF  HIGHER  ORDER 

1.  y'"  -  3  y"  =  0.     Ans.    2/  =  Ci  +  Cox  +  €36^". 

2.  y'"  —  y"  —  4  y'  +  iy  =  0.     Ans.    y  —  Cie^  +  CzC^  +  Cse"-*. 

3.  y'"  —  I6y  =  0.     Ans.    y  =  Cic^^  +  Coe"-^  +  C3  cos  2  x  +  C4  sin  2  x. 

4.  2/iv  _  6  2/"  +  9  =  0.     ^ns.   y  =  e»v/3(Ci  +  C2X)  +  e-'-^^Cs  +  C4X). 

5.  y  +  6y"'  +9y'  =  0. 

Ans.   y  =  Ci  +  (C2  +  Csx)  cos  V3  x  +  (C4  +  Csx)  sinVS  x. 

6.  r' -  16  2/'"  +  64  2/ =  0,    A;  =  2,    2,    -  1  ±_J\/3,    -liiVs: 
Ans.  y  =  e'-^(ci  +  Cox)  +  e-^[(c3+  C4X)  cosVS  x  +  (C5  +  c^x)  sin  v/3  x]. 

7.  ?/"  —  5  2/' +  4  2/ =  e^x.     Ans.   y  =zCie' —  (l/2)e'^'^+Coe'^''. 

8.  3  2/"  +  4  2/'  +  y  =  sin  x.  10.    ?/'"  —  ?/"  —  4  2/'  +  4  ?/  =  e*. 

9.  y"' —  ?jy"  +  2y' =  X.  11.   y^"  —  5y"  +  4  y  =  e"^ . 

12.  Solve  the  equation  ?/'"  +  2/'  =  0  by  first  setting  y'  =  p. 

13.  Solve  the  following  equations  by  setting  y'  —p  or  else  y"  =  g. 
(a)  3  2/'"  -  4  2/"  +  2/'  =  0.  (d)  y'"  +  3  y"  +  2  y'  =  e^. 
(/>)  2/'"  +  2/"  +  2/'  =  0.  (e)  2/i^  -  2/"  =  0. 

(c)   2/'"  +  2/'  =  sin  a;.  (/)  f"  +  v"  =  e^- 

14.  The  following  equations,  though  not  linear,  may  be  solved  by  first 
setting  y'  =p  or  y"  =  q  or  y'"  —  r. 

(a)  y'  =  y"  +  Vl  +  2/"^.  (c)   1  +  x  +  xY"  =  0. 

(b)  y" +  y"'x={y"yx*.  {d)  xy^"  +  y'"  =x^. 

15.  Solve  the  equation  x'^j/"  +  xy'  —  y  —  log  x. 
[Hint.    Put  x  =  e^  \  then 

dy  ^  dv  .  (Jz^  ^l  dy  .      d^_d  n  dv\    _  dz  ^  1  UVy      <?2/\  ■ 
da;      dz      dx     a;  dz  '      da;2     dz  \a;  dz  /      da;      x'^\dz'^      dzj' 
«o  that  the  transformed  equation  is 

yI  —  2/  =  2,  whence  y  =  c^e*  +  026-*  —  2  =  CiX  +  C2X-1  —  log  x.] 


X,  §  197]  SYSTEMS  OF   EQUATIONS  379 

16.    Solve  the  equations, 

(a)  .'•-</''  -  .o/  -3y  =  0.  (h)  xy"  -  ?/'  =  log x. 

(c)  (X  +  1)-  y"  -  4(x  +  1)!/'  +02/  =  x,   (.c  +  1  =  e'). 

(d)  (rt  +  5.»-)-2/"  +  (a  +  ^3-)^'  -y  =  log  («  +  te),   C<f  +  ^x  =  e'). 

(e)  a-V"  -(iy  =  l+x. 

195.  Systems  of  Differential  Equations.  Let  us  finally  con- 
sider systems  of  two  equations,  and  let  us  siippose  the  equations 
to  be  linear  in  the  derivatives,  that  is,  to  involve  only  the  first 
powers  of  these  derivatives. 

196.  Linear  System  of  the  First  Order.    Let  the  equations  be 

(1)  y'  =  ax  +  hij  +  cz  +  (/, 

(2)  2'  =  a^x  +  &1.V  +  t-i2  +  f?i, 

where  the  coefficients  are  constant.  We. wish  to  determine  y 
and  z  as  functions  of  x. 

Differentiating  (1)  with  respect  to  x  gives 

(3)  2/"  =  a  +  hy^  +  cz' ; 

then  the  elimination  of  z  and  z'  between  the  three  equations 
(1),  (2),  (3),  gives  a  differential  equation  of  the  second  order 
in  2/,  which  should  be  solved  for  y. 

197.  dx/P  =  dy/Q  =  dz/R.  Here  P,  Q,  and  R  are  functions 
of  X,  y,  z.  Let  X,  /x,  v  be  any  multipliers,  either  constants  or 
functions  of  x,  y,  z.     Then,  by  the  laws  of  algebra, 

,^.  dx  _  dy  _dz  _  \dx  4-  fxdy  +  vdz 

^  ^  ~P~'  Q~B~  XP  +  fjiQ+vE' 

Suppose  that  we  can  select  from  these  ratios  (or  from  these 
together  with  others  obtainable  from  them  by  giving  suitable 
values  to  X,  fi,  v)  two  equal  ratios  free  from  z,  i.e.  containing 
only  X  and  y.  Such  an  equation  is  an  ordinary  differential 
equation  of  the  first  order  in  x  and  y.     Solving  it,  we  obtain 

(2)  f(x,y,c,)=0. 

Suppose  that  a  second  pair  of  ratios  can  be  found,  free  from 


380  DIFFERENTIAL   EQUATIONS  [X,  §  197 

another  of  the  variables,  say  y.     The  result  is  an  equation  of 
the  first  order  in  x  and  z.     Let  its  solution  be 
(3)  Fix,  z,  c,)=  0. 

Then  (2)  and  (3)  form  the  complete  solution  of  the  system. 
Conversely,  differentiating  (2)  and  (3)  with  respect  to  x,  elimi- 
nating Ci  and  Cg,  and  solving  for  dx:dy:  dz,  we  find  a  system 
like  (1).  In  selecting  the  second  pair  of  ratios,  the  result  (2) 
of  the  first  integration  may  be  utilized  to  eliminate  the  variable 
whose  absence  is  desired. 

Example  1.  dx/a-2  =  dy/xy  =  dz/z^. 

The  first  two  ratios  give  dx/x  =  dy/y,  whence  y  =  ciX.  Putting  this  value 
of  y  in  dy/xy  —  dz/z"^  gives  dy/^ciy"^)  =  dz/z^,  so  that 

ciy     z 

or,  z  —  C\y-\-  C\Ciyz  =  x  +  c^xz.    Hence 
the  solutions  are    given  by  the  two 
equations  y  —  CiX,  z  =  a;  +  c^xz. 
Az 


Interpreted  geometrically,  the  solu- 
^  tions  represent  a  family  of  planes  and 

a  family  of  hyperboloids.  These  are 
the  integral  surfaces  of  the  differen- 
tial equation.  Each  plane  cuts  each 
hyperboloid  in  a  space  curve,  forming 
Fig.  76  a  doubly  infinite  system  of  curves,  the 

integral  curves  of  the  differential 
equation.  The  system  may  be  written  dx  -.dy  :dz  =  x^  :xy  :  y^.  But  the 
direction  cosines  of  the  tangent  to  a  space  curve  are  proportional  to  dx, 
dy,  dz.  Thus  the  given  equations  define  at  each  point  a  direction  whose 
cosines  are  proportional  to  x~,  xy,  y'^.  Our  solution  is  a  system  of  curves 
having  at  each  point  the  proper  direction.  What  curve  of  the  above  sys- 
tem goes  through  (4,  2,  8)  ?  What  are  the  angles  which  the  tangent  to 
the  curve  at  this  point  makes  with  the  coordinate  axes  ? 

Example  2.  _d^ ^ _dy_ ^ ^z_ ^ 

y  —  z     z  —  X     X  —  y 
Let  \=  n=  v=\.     Then  each  of  the  above  fractions  equals 
dx  +  dy  +  dz 
0 


X,  §  197]  SYSTEMS  OF  EQUATIONS  381 

But  since  the  given  ratios  are  in  general  finite,  this  gives 
dx  -{-  dy  +  dz  =  0,     vrhence    x  +  y  +  z  =  Ci. 

Again,  let  X  =  x,  m  =  2/1  v  =  z-     This  gives 

xdx  +  ydy  +  zdz  =  0,    whence  x'-^  +  y-+  2-  =  C2. 

Thus  the  integral  surfaces  are  planes  and  spheres,  and  the  integral 
;urves  are  the  circles  in  which  they  intersect. 

In  this  example  the  multipliers  \,  /a,  v  have  heen  chosen  so  as  to  get 

;xact  differentials. 

Examples.  _d^^_diL_  =  ^. 

x  —  y     x+  y      z 

rhe  first  two  ratios  are  free  from  z  and  give 


arc  tan  (y/x)  =  log  [cix'V  v  x^  +  y-\ 

Losing  the  multipliers  \  =  x,  m  =  2/i  k  =  0,  and  equating  the  ratio  thus 
abtained  to  the  last  of  the  given  ratios,  we  find 

^^^y^y  =  ^ ,     whence  x'^  +  y^  =  c^^^. 
x2  +  y^        z 

EXERCISES  LXXXI.  —  SYSTEMS  OF  EQUATIONS 

1.  xdx/y^  =  ydy/x^  =  dz/z.  Ans.    x*  -y*  =  Ci;  z- =  C2(x-  +  y-)- 

2.  dx/x  =  dy/y  =  -  dz/z.  A71S.    yz  =  Ci;  y  =  CoX. 

3.  dx/yz  =  dy/xz  =  dz/(x +  y). 

Ans.   z-  =  2(x  +  y)  +ci;  x'^-y'  =  co. 

4.  dx/(y  +  2)  =  dy/(x  +  z)  =  dz/(x  +  y). 

Ans.    (x-y)  =ci(x-z)  =C2(y-z). 

5.  dr/(x2  +  2/2)  =  dy/(2  xy)  =  dz/(xz  +  yz). 

Ans.   2  y  =  ci{x^  -  y-)  ;  x  +  y  =  C2Z. 

6    <^y  -        ^^y  (Z^  _        2xz 

dx~x2-  J/2_22'     (Ix        j2_yi_zl 

Ans.   y  =  riz  =  C2(x2  +  j/2  4.  z-) . 

„    dy  _  z  —  Sx  .   dz  _  2x  —  y 
'  dx     3  y  —  2  z'   dx     Sy  —  2z 

Ans.    x  +  2y  +  Szz=ci;  x"^  +  y'^  +  z- =  Co. 

8.  dx  z=  —  kydt ;  dy  =  kxdt. 

Ans.   X  =  A  CO?,  kt -It  B  sin  kt\  y  =  A  sin  kt  -  B  cos  kt. 

9.  dx/dt  =  Sx  —  y  ;  dy/dt  =  x  +  y. 

Ans.  x=(^A  +  Bt)e^;  y  =  (^  -  B  +  50«^. 


382  DIFFERENTIAL   EQUATIONS  [X,  §  197 

10.  Determine  the  curves  in  which  the  direction  cosines  of  the  tan- 
gent are  respectively  proportional  to  the  coordinates  of  the  point  of  con 
tact ;  to  the  squares  of  those  coordinates. 

11.  A  particle  moves  in  a  plane  so  that  the  sum  of  the  axial  compo- 
nents of  the  speed  always  equals  the  sum  of  the  coordinates  of  the  parti- 
cle, while  the  difference  of  the  components  is  a  constant  k.  Determine 
the  possible  paths.  Ans.   x  +  y  =  Cie' ;  x  —  y  =  kt  +  Co. 

12.  If  the  particle  in  Exercise  11  is  at  (1,  1)  when  t  —  0,  where  is  it 
when  t  =  5?    Approximately  how  far  has  it  traveled ? 

198.  Partial  Differential  Equations.  While  a  general  treat- 
ment of  differential  equations  which  involve  partial  deriva- 
tives is  beyond  the  scope  of  this  book,  a  few  examples  that 
can  be  solved  without  special  theory  will  illustrate  the  nature 
of  such  equations  and  their  solutions. 

Example  1.  Solve  the  equation  dz/dx  =  0,  where  z  is  some  function 
of  X  and  y. 

Since  dz/dx  =  0,  we  have  z  =  "  const."  —  in  so  far  as  x  is  concerned. 
But  during  a  partial  differentiation  with  respect  to  x,  y  acts  like  a  con- 
stant ;  hence  any  arbitrary  function  of  y,  A(y),  may  be  put  in  place  of 
the  constant  of  integration,  and  the  general  solution  \s:  z  —  A(y). 

Example  2.     Solve  the  equation  dz/dx  —  2x  +  y. 

Since  y  may  be  thought  of  as  a  constant  during  the  integration  with 
respect  to  x,  we  may  integrate  at  once  term  by  term,  thinking  of  y  as  a 
constant :  z  =  x'^  +  xy  +  A(y),  where  the  arbitrary  function  of  y,  A(y)t 
takes  the  place  of  the  usual  constant  of  integration,  as  in  Ex.  1. 

Example  3.     Solve  the  equation  d'^z/dx  dy  =  2x  +  y. 

Integrating  first  with  respect  to  x,  we  find,  by  a  repetition  of  the  work  - 
of  Ex.  2,  dz/dy  =  x-  +  zy  +  A{y).  Integrating  again,  this  time  with  re- 
spect to  y,  thinking  of  x  as  constant,  we  find 


^^y+^  +  ^A{y)dy  +  B(x), 


where  B(x)  is  any  arbitrary  function  of  x,  —  a  constant  with  respect  to 
y.  Since  fA{y)dy  is  itself  completely  arbitrary,  the  general  solution  may 
be  written 

z^xhj  +  '^+fix)  +0(2/), 
where /(x)  and  ^(j/)  are  arbitrary  functions  of  x  and  y. 


X,  §  199]  PARTIAL   DERIVATIVES  383 

199.  Relation  to  Systems  of  Ordinary  Equations.  It  is 
shown  in  works  on  Differential  Equations  that  the  linear  par- 
tial differential  equation  of  the  first  order : 

(1)         p{x,  2/,  ^)  ^  +  Q  {x,  y^^)Yy  =  ^  (^^  y^  '\ 

can  be  solved  if  the  system  of  ordinary  equations  (§§  196-197) 
^2)  f^^       _        dy        _        dz 


P{x,y,z)  Q(:x,y,z)  R{x,y,zy 
can  be  solved.  In  fact,  if  the  solutions  of  (2)  can  be  written 
in  the  form  f{x,  y,  z)  =  const.,  <^{x,  y,  z)  =  const.,  then  the 
general  solution  of  (1)  is  f(x,  y,  z)  =  A  [<f}{x,  y,  z)],  where  A 
denotes,  as  in  §  198,  an  arbitrary  function.  The  truth  of  this 
fact  can  be  established  by  a  direct  check  in  equation  (1). 

Example  1.     Solve  the  equation  x'^(dz/cx)  +  xy(dz/di/)  =  z^. 

The  auxiliary  equations  (2)  above  become  dx/x^  =  dy/{xy)  =dz/z'^. 
These  were  solved  in  Ex.  1,  §  197  ;  the  solutions  may  be  written :  y/x  — 
Ci,  (z  —  x)/{xz)  =  c-,.  Hence  the  general  solution  of  the  given  equation 
is  (s  -  x^lixz)  —  A{y/x)  or  z  =  x  +  xz  A{y/x). 

EXERCISES  LXXXII— PARTIAL  DIFFERENTIAL  EQUATIONS 

I        1.   Solve  each  of  the  following  equations,  where  z  represents  a  function 
'   of  X  and  y  -. 

I       (a)l^  =  c.  id)'^  =  x^--y^.  ^9)^  =  %. 

ex  Bx  dxdy     dx 

*^  ?2^  ?2r  P^Sv 

^  '   cx^  ^       dxdy  dx^cy 

(c)      '''  -0  (f)  '^-2x  (i)   ^!^-^ 

■       ^'^    illy- ^-  ^-^^  cy^ -     ""■  ^^    exit  ~ dy-' ■ 

2.  Solve  each  of  the  following  equations  by  comparison  with  a  cor- 
responding example  (see  §  199)  of  List  LXXXI. 

(a)   yJ^^  +  t^^  =  ,.  (c)    zy^^  +  xz^-'  =  x  +  y. 

X  dx      y  cy  ex  dy 

(^b)   x^+y'^  +  z  =  0.  (d)    (y  +  z)f--\-(.x  +  z)^^=x  +  y. 

dx         dy  dx  cy 


INDEX  TO   TABLES 

References  to  pages  of  the  Tables  in  Italic  numerals. 
References  to  pages  of  the  body  of  the  book  in  Roman  numerals. 

PAGE8 

Table      I.  Signs  and  Abbreviations 1-3 

Table     II.  Standard  Formdlas 3-16 

Table  III.  Standard  Curves 17-32 

Table  IV.  Standard  Integrals 33-48 

Table     V.  Numerical  Tables 49-68 


Greek 

Alphabet 

Letters    Names 

Letters      Names 

Letters      Names 

Letters 

Names 

A  a      Alpha 

H,; 

Eta 

N  V      Nu 

Tr 

Tau 

B  ^     Beta 

e  e 

Theta 

Ss^       Xi 

Ti; 

Upsilon 

r  7     Gamma 

It 

lota 

0  0      Omicron 

$0 

Phi 

A  5      Delta 

Kk 

Kappa 

Htt     Pi 

Xx 

Chi 

E  e      Epsilon 

AX 

Lambda 

Pp      Rho 

^i. 

Psi 

Z  f      Zeta 

Mm 

Mu 

S  (T  J  Sigma 

fi  w 

Omega 

TABLES 


[Roman  page  numbers  refer  to  the  body  of  the  text ;  italic  page  numbers  refer  to  these 
Tables.] 

TABLE  I 
SIGNS  AND   ABBREVIATIONS 

1.  Elementary  signs  assumed  known  icithout  explanation  : 

+  ;    ±  ;    T;    —  ;    =;    a  y.  b  =  a  ■  b  —  ab  ;    a  -i-  b  =  a/b  =  a  -.b  =  -; 

b 

cfi;    a^iffl";    a-"  =  l/a!";   ai/"  =  \/a ;    aP'^^Va^;    a^  =  l;    ();    []; 

\    \  ;  ;   «',  rt",  •••,  a(")  (accents)  ;   ai,  a^i  •••,  a„  (subscripts). 

2.  Other  elementary  signs  : 

^ ,  not  equal  to.  >,  greater  than  or  equal  to. 

>,  greater  than.  <,  less  than  or  etjual  to. 

<,  less  than.  n!  (or [n), factorial  n  =  n(n— 1)  •••  3  •  2  •  1. 

q.p.,  approximately.  |  a  |,  absolute  or  numerical  value  of  a. 

3.  Signs  peculiar  to  The  Calculus  and  its  Applications : 

(a)   Given  a  plane  curve  y  =f(x)  in  rectangular  codrdinates  (.r,  y)  ; 
Hi  =  slope  =  dy/dx  —  f  (x)  =  y'  =  first  derivative  ;  see  p.  23. 
[Also  occasionally  D^y,  f^,  y,  p,  by  some  writers.] 
a  =  angle  between  positive  a;-axis  and  curve  =  tan-^  ni. 
Ay,  A-y,  •••,  A">/,  first,  second,  •••,  n"'  differences  (or  increments)  of  y. 
dy  =f(x)  •  Ax,  d-y  =/"(x)  •  A?,  ••-,  d^y  =f('0(x)  •  Ax",  first,  second, 
•••,  ?i"i  differentials  of  y. 

rr  =  relative  rate  of  increase,  or  logarithmic  derivative ;  see  p.  146  ; 

=  /(^)  -/(^)  =  {(ly/dx)  -  y  =  d  {log  y)/dx  =  r^  -f-  100. 
Vp  =  percentage  rate  of  increase  =  100  •  r,. 

b  =  flexion  =  d-y/dx^  =/"(.r)  =  ?/"  =  second  derivative  ;  see  p.  71. 
d'^y/d.f"  =/<")(x)  =  y("'>  =  n^^  derivative. 

K=  curvature  —  l-i-B;  R  =  radius  of  curvature  =  1  -^  A';  p.  170. 
1 


SIGNS  AND  ABBREVIATIONS  [1,3 

J"/(x)  dx  =  indefinite  integral  of /(.r);  see  p.  96. 
\  f(x)  dz=  \       f(x)  dx  =  definite  integral  off(x);  see  p.  115. 
s  =  length  of  arc  ;   s  \        =  arc  between  x  =  a  and  x  —  b. 

:=6 

=  area  between  y  =0,  y-f(x),  x  =  a,x  =  b;  see  p.  116. 


o:-]; 


(6)   Given  a  curve  p  =f(0)  in  polar  coordinates  (p,  0)  : 
^j/  —  Z  (radius  vector  and  curve)  =  ctn-i  [(dp/dd)  -=-  p] 
=  ctn-i  [d  (log  p)/de']. 
<p  =  Z  (circle  about  0  and  curve)  =  tan-i  \^(^(ip/(id)  -^  p] 
=  tan-i[d(logp)/d!6']. 
^      =  ^  =  area  between  p  =f(0),  e  =  a,  6  =  p  ;  see  p.  212. 

(c)  For  problems  in  pla^ie  motion: 

s  =  distance.  Vx  =  horizontal  speed  =  projection  of  v  on  Ox. 

t  =  time.  Vy  =  vertical  speed  =  projection  of  v  on  Of/. 

m  =  mass.  jx  =  horizontal  acceleration  =  proj.  ofj  on  Ox. 

V  =  speed.  jy  =  vertical  acceleration  =  proj.  of  J  on  Oy. 

V  =  velocity  (vector).  j,v  =  normal  ace.  =  proj.  of  j  on  the  normal. 
j  =  ace.  (vector) .  jr  =  tangential  ace.  =  proj.  of  j  on  the  tangent. 

0  =  angle  (of  rotation),    a  =  angular  acceleration. 

a>  =  angular  speed.  g  =  acceleration  due  to  gravity. 

(d)  Problems  in  space;  functions  z  =f(x,  y,  ••■)  of  several  variables  : 
Previous  notations  are  generalized  when  possible  without  ambiguity, 


exceptions  are 


p  =  dx/dx  =fx;     q  =  dz/cy  =  fy  ; 


r  =  d'^zlcx?  =  /„  ;     s  =  cHjdx  dy  =  f,y  ^fy^;     t  =  d^ldf-  =  /,„. 
[The  notation  {dz/dx),  used  by  some  writers  for  dz/dx  is  ambiguous.] 

4     Other  letters  commonly  used. with  special  meanings : 

If  =:  ratio  of  circumference  to  diameter  of  circle  =  3.14159---. 

e  =  base  of  Napierian  (or  hyperbolic)  logarithms  =  2.71828---. 

M=  logio  e  =  modulus  of  Napierian  to  common  logarithms  =  0.434- •■ 

^  =  "  sum  of  such  term  as  "  ;  thus  :   ^'"a;^  =  «i"  -\-  a.?  +  ■■■  aj. 

(a,  p,  7),  —  direction  angles  of  a  line  in  space. 

(?,  m,  n),  —  direction  cosines  ;  I  =  cos  «,  etc. 

S.  H.  M.  — simple  harmonic  motion. 

e  or  e,  — eccentricity  of  a  conic  ;  also  phase  angle  of  a  S.  H.  M. 


II,  AJ  EXPONENTS  AND   LOGARITHMS  3 

a,  —  amplitude  of  a  S.  H.  M. 

(rt,  b),  — semiaxes  of  a  conic  ;  (a,  b,  c),  semiaxes  of  a  conicoid. 

A  =  difference  (of  two  values  of  a  quantity). 

p  —  density  ;  also  radius  vector,  radius  of  curvature,  radius  of  gyration. 

5.  Trigonometric,  logarithmic,  hyperbolic,  and  other  transcendental 
functions :  See  Tables,  II,  A  ;  II,  F,  3  ;  II,  G  ;  II,  H  ;  and  consult  Index. 

6.  Inverse  function  notations  : 

If  y  =f(x),  then  f~^{y)  =  x;  f-^  denotes  an  inverse  function.  [This 
notation  is  ambiguous  ;  confusion  with  {/(x)}"^  =  1  -^f{x).'] 

sin-i  X  or  arc  sin  x,  —  inverse  of  sin  x,  or  anti-sine  of  x,  or  arc  sine  x, 
or  angle  whose  sine  is  x.  [Other  inverse  trigonometric  functions,  and 
hyperbolic  functions,  follow  the  same  notations.  See  Tables,  II,  G,  18 ; 
H,7.] 

TABLE  II 

STANDARD   FORMULAS 

A.     Exponents  and  Logarithms. 

(The  letters  B,  b,  etc.  indicate  base;  L,  I,  •■•  indicate  logarithm;  JV,  », 
•  ••  indicate  member ;  base  arbitrary  when  not  stated.     See  §  73,  p.  130.) 

Laws  of  Exponents  Rules  or  Logarithms 

(1)  N  =  BL;  in  particular  (1)'  L  =  logeN,  i.e.  iV^=  B'^es^;  and: 
1=50;  B=Bi;  \/B=  E'K  logl  =  0;  log2,5=];  log|,(l/5)=-l. 

(2)  B^B'  =  B^+i.  (2)'  log  (iV .  n)  =  log  .V  +  log  n. 

(3)  Bi-^  Bi  =  B^-i.  (3)'  log  (.V-f-  n)  =  log  X-  logji. 

(4)  {B'-Y  =  B^i-.  (4)'  log  (i\r»)  =  71  log  X. 

(5)  N  =BL,  B  =  fe*,  X  =  6*i.  (5)'  logfc  X  =  log6  B  .  log^  .V. 

B=e,  6  =  10  gives  *=0.4.34294r}=  J/"=logioe  ;       \ogiay=Jf-  loge  X 
5=10,  6=e  gives  *=2.302585=l-=-Ji/=lope10;  log,  3^=(l-=-J/')  log,oxV. 
6=JV  gives  L=]A,  l=log65.  logjft;      e.(7.,  log«10=l-=-log,oe. 

L=x  gives  10*=«»-^-«';  «z=io*x. 

2f=x  gives  logger  =  3/' •  log,  a- ;  log,  ir=(l  + JT)  log,oa!. 

(6)  y  =  ex"  gives  v  =  nu  +  k,  Ji  =  logm  x,  v  =  logio  y,  k  =  logio  c. 
(j)  y  =  ce"   gives   v  =  mx  -\-k,   v  =  logi,,?/,    m  =  a  logio  e  =  aM, 
k  =  \ogioc. 


4  STANDARD  FORMULAS  [II,  B 

B.  Factors. 

(1)  a2  -  62  =  (a  _  5)(a  +  5).  (2)   (a  ±  6)2  =  a^±2ab  +  b^. 

(3)  a"-  ft-  =  (a  -  6)(a'»-i  +  a"-^  6  +  a'^-^b^  +  ■■■  +  6»-i). 

(4)  a2n+l  +  52n+l  =  („  +  5)((i2n  _  (^2n-lft  +  ...   +  62») , 

See  also  Tables,  IV,  Nos.  16,  20,  21,  49,  50. 

(5)  Polynomials :   if  /(a)  =  0,  /(x)  has  a  factor  x  —  a;   in  general : 
/(x)  -=-  (x  —  a)  gives  remainder  /(a). 

(6)  (a  ±  6)"  =  a»  ±^a''-^b  +  "^^'^^  a"-262+  ...  +  (±l)»6n. 
See  II,  B,  1,  p.  7.  ^  ^  '  ^ 

C.  Solution  of  Equations. 

(1)  ax2  +  6x  +  c  =  0,  roots:    x  —  A  ±  ^ft^jzl^^  =  _  A  ±  V^^, 
2a  2a  2a      2a 

where 


f  real 
D  =  b"  —  iac;  roots  of  (1)  are   ]  coincident 


>0 

when  D  \  =0 

<0 


[  imaginary 

(2)  X"  + /»i.r"-i  +  pax"-!  +  •• +i)„-ix +p„  =  0.  Roots  :  Xi,  Xo,  •■•,  x„ ; 
then  ^Xj  =  — i^i,   X^«3^y  =i'2i  ^x,x,Xt  =  — ps,  etc. 

(3)  /(x)  — 0(x)=:O:  roots  given  by  intersections  of  y  =/(x),  y=<p(x). 
(Logarithmic  chart  often  useful. )  Find  roots  approximately  ;  redraw 
figure  on  larger  scale  near  intersection.     {Oeneralized  Horner  Process.) 

(4)  Simultaneous  Equations  :  /(x,  y)  =  0,  0(x,  y)  =  0  :  roots  (x,  y) 
are  points  of  intersection ;  redraw  on  larger  scale  as  in  (3). 

(5)  Linear  Equations  : 

(a)  2  equations  in  2  unknowns :         ^  ly  —   i 

1  aox  +  622/  =  C2 

Solutions  :  X  =  I  ^1^^  I  -  I  "1^1 1  =  (C162  -  C261)  -4-  (ai62  -  a2&i), 

I  (^2^2  I         I  fl2&2  I 

y  =  I  "'^'  I  -  i  '^'^  I  =  (aiC2  -  asci)  -  (a,62  -  ^261). 

I  a2C2  I         I  0202  I 


II,  D] 


FORMULAS  OF  ALGEBRA 


(b)  71  equations  in  n  unknowns  :   a.JCi  +  biX-2  +  •••  +  k,x„  =  Pi  ;  i  =  1, 
2,  ...,M. 

pi  ■■•  A-i 


Solutions  :   x,  = 


where 


a.,  h., 


/),...  A-. 


an6„ 


^D     \ 


I  Column  of  j9's  replaces  column  of  \ 


coefficients  otssi. 


aibi- 

■h 

b.. 

■ko 

6i- 

■ki 

a-2  bo  • 

•  A-o 

=  ffl 

b,. 

•A-3 

-a-z 

b,. 

■ks 

OnK- 

•*„ 

K- 

•A-„ 

bn- 

■K 

+  ... +(_l)»-ia„ 


fci-A-i 
b^-k. 


K. 


[Coefficient  of  «,-  skips  2th  row  of  D.     The  last  formula  is  a  general 
aefinition  of  a  determinant.] 

D.    Applications  of  Algebra. 

1.  Interest.     (P  =  principal ;   p  =  rate  per  cent ;    r  =  p  h-  100 ;    n  = 
number  of  years  ;  An  =  amount  after  n  years.) 

(a)  Simple  interest :  ^„  =  P(l  +  nr). 

(b)  Yearly  compound  interest:  A„  =  P ■  (1  +  r)". 

(c)  Semiannually  compounded  :  A^  =  P{1  +  r/'I)'^^. 

(d)  Compounded  once  each  ?Hth  part  of  year  :  J„  =  P(l  +  r/m)'^". 

(e)  Continuously  compounded  :  A„—  P  lim  (1  +  r/m)"*"  =  Pe"'. 

2.  Annuities.    Depreciation.     (/=  yearly  income  (or  depreciation  or 
payment  or  charge)  ;  7i  =  number  of  years  annuity,  or  depreciation,  runs.) 

(«)  Present  worth  P  of  yearly  annuity  /: 

P=7[(l  +  r)n-l]-^[r(l  +  r)»J. 


(b)  Annuity  /purchasable  by  present  amount  P;  or,  yearly  deprecia- 
tion /  of  plant  of  value  P  : 

7=P[r(l  +  r)'']^[(l  +  r)'«-l]. 

(c)  Final  value  An  of  n  yearly  payments : 

^  =  /(l  +  r)[(l  +  r)»-l]^r. 


6  STANDARD  FORMULAS  [II,  D 

3.  Permutations  Pn,ri  (t'ld  Combinations  C„^r,  of  n  things  r  at  a  time, 
witliout  repetitions : 

(a)  Fn,r  =  n{n-  1)  •■•  {n  -  r  +  l)=n!  -(n-r)! 

(b)  C„,r  =  Pn,r-^r\  =[«(9i-l)  •••  (?i -  r  +  1)] -J-  f ! 

4.  Chance  and  Probability. 

(a)  Ctiance  of  an  event  =  (number  of  favorable  cases)  h-  (total  number 
of  trials)  <1. 

Chance  of  successive  (independent)  events  =  product  of  separate  chances  <1. 
Chance  of  at  least  one  of  several  (independent)  events  =  sum  of  separate  chances. 

(b)  Probable  value  v  of  an  observed  quantity : 

V  =  (  ^ mi  ]-7-  n=  arithmetic    mean    of    n   measurements  mi,    m^,   •••, 
m„  ;  probable  error  in  w  =  ±  .6745  "V  (  ^  (■"  —  "*0^  j  -=-  ?i  (k  —  1). 

(If  the  observations  are  unequally  reliable,  count  each  one  a  number  of  times,  p^,  which 
represents  its  estimated  reliability  ;  Pi  =  "  weight "  of  ?«^). 

(c)  Probable  value  of  k  in  formula  v=:  kx  : 

A;  =  ^  XiVi  -7-  ^-^'i^'  ^^^^  ""-  measurements  (xi,  Ui),  (xo,  vo),  .••,  (a;„,  v„) ; 

probable  error  in  A;  =  ±  .6745  -W^  {kZi  —  Vi)"^  -f-  (n  —  l)"V;ri2.    See  Ex.s. 
4,  p.  262  ;  29,  p.  342.  ^^ 

(d)  Probable  values  of  k,  I,  m,  •••,  in  formula  v  =  kz  +  hj  +  mz  +  ••• 
are  solutions  of  the  equations : 

k  ^^'i^  +  I  ^  Xiiii  +  m  ^  XiZi  +  •••  =  "^XiVj 
k  ^.r^yi  +  I   2j?/i2  +  m  ^  yiZi  +  .••  =  ^yiVi 

k  ^Xj^i  +  I     2  2/>'^«  +  *"   2^'^  "*"    ■■■    ~     Z^^i'^i 


See  also  Exs.  18-23,  p.  69,  Example  2,  p.  323,  and  Exs.  10-22,  p.  328. 
{Rules  for  Least  Squares.     See  also  Observational  Errors,  No.  Ill,  J.) 


II,  E]  SERIES  7 

E.    Series. 

1.   Binomial  Theorem :    Expansion  of  {a  +  b)". 

(a)  n  a  positive  integer  :  (a  +  6)"  =  0"+  2"^"  Cn,ra"~^b'' ; 

[r,i,r:  see  No.  II,  D,  3,  p.  6,  and  also  II,  B,  6,  p.  U.] 

(ft)  n  fractional  or  negative,  |  a  |  >  |  6  | : 
(a  +  6)"=a»  4-  ^a"-^&  +  "^'^~  ^^a"-^&'H  •••  +  C„,r«"-''6''+  •••  (forever), 
(c)  Special  cases : 

— ! —  =(1  ±  j)-!  =  1  T  a-  +  ■'•=  T  ■T^+  a^  T  ■•• ;  (  I  •'•  I  <  !)•     (Geometric  progression.) 
l±a- 

Vr^^  =(1  ±  ,r)^/2  =  1  ±  l.r  -  _l_.r2  ±  -i^.r^  _...;(!  ^  |<  1). 


vrr^ 


92 .  -^ : 


2.  Arithmetic  series  :  n +(a  +  d)  +  (a  +  2  d)+  ■■■  -[-{a +(n—  l)d); 
last  term  =  1  =  a  +(n  —  l)d ;  sum  =  s  =  n(a  +  l)/2. 

3.  Geometric  series  :  a  +  ar  +  ar-  +  nr^  +  •••. 

(a)  n  terms  :  Z  =  ar'^-^  ;  s  = =  a • 

?•  —  1  r  —\ 

(6)  infinite  series,  |  r  |  <  1  :  s  =  a/(l  -  r). 

4.  1  +  2  +  3  +  4+  -  +(H-l)+?i  =  h(h  +  1)/2. 

5.  2  +  4  +  6  +  8  +  •••  +  (2  ?«  -  2)  +  2  H  =  ?i(«  +  1). 

6.  1  +  3  +  5  +  7  +  ■••  +(2;i-3)  +  (2?j-l)  ^  n\ 

7.  12  +  22  +  32+  ...  +(«  -  l)2  +  n2  =  ,j(„  +  l)(2  7i+  l)-3! 

8.  13  +  2^+33+  ...  +(n-  1)3 +  k5  =  [„(„+  l)/2]2. 

9.  1  +  1/1!+  1/2!  +  1/3!+  .-  =lim^l+-y  =  c  =  2.71828.... 

10.  <"  =  1  +  x/1 !  +  a;V2  '•  +  a;'/-"^ !  •••  ;  (all  x) ;  a'  =  e^'-'g". 

11.  log,(l  ±x)  =  ±a;-x2/2±x3/3-x^/4±rV5 ;  (-l<J-<  +  l). 

12.  log.[(l+x)/(l-x)]=2[a;  +  xV3  +  a;V5+  -];  (-l<x<  +  l). 
fCoinputalionoflogJV:  set  iV=(l  +  x)/(l  -a-);  then  e  =(^- l)AxV+ 1);  use  II,  A,  5'.] 


8  STANDARD  FORMULAS  [II,  E 

13.  sin  X  =  x/1  !  -  x^S  !  +  x^/5  !  -  xV?  !  +  -•.  ;  (all  x). 

14.  cos  x  =  l-  xV2  !  +  ar'/4  \  -  7^/6  1 +- ;  (all  x). 

15.  tan  x  =  x  +  a^/S  +  2  x^lb  +  17  x^Slb  +•••;(  I  « |<  t/2). 
General  term  :  22»  (2^"  -  l)£2n-l  -^(2»)! ;  see  ^„,  TaftZes,  V,  N,  p.  58. 

16.  ctu  x=  1/x  -  x/S  -  xV45  - ^2  52«-i(2 x)2»-i-  (2  n) ! ; 

(0<|x|<7r). 

17.  sec  X  =  1  +  xV2  !  +  5  x^/i  !  +  2j  [-B2n:«2V(2  «)!];(  I  ^  l<  'r/2). 

18.  CSC x=  l/x+x/3!  +  V  [2(22«+i-  I)52«4.ix2"+V(2  n+2)!] ; 

(0<|x|<7r). 

19.  sin-ix  =  7r/2-cos-ix=x+xV(2-3)  +  1.3xV(2-4.5)+.-.;(|x|<l). 

20.  tan-i  X  =  7r/2  -  ctn-i  x  =  x  -  x^/3  +  x^/b -  xyi  +  ••■;  ( |  x  |  <  1). 

21.  (e^  +  e-^)/2  =  cosh  x  =  1  +  x2/2  !  +  x4/4  !  +  x^/6 !  +  -•  ;  (all  x). 

22.  (e^  -  e-^)/2  =  sinh  x  =  x  +  xV3 !  +  x^/6 !  -f  xV?  !  +  •••;  (all  x). 

23.  e-*'  =  1  -  x2  +  xV2 !  -  x6/3 !  +  xV4 ! ;  (all  x). 

24.  /(x)  =:/(«)  +/(a)(x  -  a)  +/"(«)  (x  -  a)2/2  1  +  ... 

+  /»-i(a)(x-  a)'-V(n  -  1)!  +E„. 

Taylor's  Theorem ;  Remainder  E„  :  \  £"„  |  <  [Max.  I/^''^-^)  |  ]  [  (a-  -  «)"  1  +  w  1 ; 
^n  =/"[a-hp{a--a)](x-a)n^n  ! ;  £"„  =(1  -/))"-l/("^[rt  +i?(a'  -  o)]  (x  -  a)"/i !; 
IPl<l. 

Set  a  =  0  :  /(x)  =/(0)  +/'(0)a-  +/"(0)a^V3  !  +  -  +/"-l(())a-"-V(ra-l)  !  +  -E'n; 

[3/acZaMWn]. 

Set  aj  =  r  +  A,  a  =  r  :  /(r  +  /()  =/(r)  +  /i/'(r)  +  h\f"(,r)/2  :  +  .••+  ^„. 

25.  /(aj  +  ^,  y  +  k)  =f(x,  y)  +  [A/c(ir,  y)  +  Xy^Ctc,  y)] 

+  [Wzz  +  2  AX;/^„  +  k\fyy\  .^  2  !  +  ...  +  ^„ ; 
I  ^„  I  <  J/C I  A  1  +  I  A;  I )"  4-  71 !,  J/  =  maximum  of  absolute  values  of  all  ft"*  derivatives. 

26.  If /(a-.)  =  fifo/2  +  fli  cos  X  4-  «2  cos  2  a;  +  Og  cos  3  a;  +  ... 

4-&1  sin  a;4-&3sin  2a5  +  53Sin  3a;H ;     {-ir<x<  +  ir). 

an=-  I  /(a;)  cos  na-.  rf^r ;  6„  =  -  I  f{ai)%\a.nxdx.     Fourier  Theorem, 

F.    Geometric  MaguitudeB.     Mensuration. 

I  =  length  (or  perimeter) ;  A  —  area  ;  V  =  volume. 


II,  F] 


MENSURATION 


Dimensions  or  Equations 


1.  Triangle. 


^ 


2.  Trapezoid. 


EV^ 


AOEB 
Z.OBT 
Z.FBT 
ZFBO 

=  a/2; 

=  90"; 

=  90=  -  a 

4.  Ellipse.    e<\. 

/'     ^ 

''^i^ 

Y^y"'' 

ilfoi^. 

%pf 

\^ 

^-dJ 

-  +  ^=1,  (originate). 


(pole  at  Jf)  ; 
Foci,  F,  F' ;  Center  ;  O. 


Sides:  rt, 6,0.  Anples:  ,4,7?.('. 
Altitude  from  A  on  «  =  /i,,. 

s  =  (a  +  6  +  c)/2  ; 

IT  =  . -1  +  .fi+  6"=  ISO"; 


=  («  -  a)  tan  {A/i) ; 
c  =  6  cos  vi  +  a  cos  JS ; 
fi  =  „s  -I-  J2  _  2  (/6  cos  f. 


h  =  height.    &i,  ftj  =  bases 


r  =  radius  ;  d  =  diameter  ; 

a  =  COB  at  centei* 

arc  CB  ,     ,.       . 
= 1 —  (radians) 

1  Sn  arc  CB  ,, 

•= (degrees) ; 

TT         r 

a./-l  =  CEB,  <^  =  2a.  =  DOB; 

sin  a  =  FB  -i-  r  =\  -i-  esc  a  ; 

cos  a  =  OF  -J-  /•  =  1  -H  sec  a  ; 

tan  a  =  r^  -7-  r  =  1  -5-  ctn  a ; 
vers  a  =  FC  -4-  r  =  1  —  cos  a  ; 
e-xseca^rrn-  r  =  seco-1. 
tan  (a/2)  =  BF^  EF=  sin  a 
sin  (o/2)  =  ^/'H-  F5  =V(T 
cos  (a/2)  ^  EF^  EB  =%/(! 


l  =  rt+«.  +  c  =  2s; 
u4  =  aAa/2  =  6/H/2  =  o;ie/2 
=  (1/2)  aft  sin  C,  etc. 
=  r«  =Vs(«-a)(.s-6)(«-c); 
sin  J.  _sin5_  sin  f 
a     ""     6     ~  "c     ' 


6  + 


■etn- 


./£  =  /.  (^1  +  hi,  /2. 


i  =  2  rrr  =  -nd  =  2  vl/r  ; 
A  =  n>i  =  7r<7V4  =  i  r/2  ; 
arc  C.S  =  r  •  a,  (a  in  radians) 

=  jrra/lSO,  (a in  degrees); 
Chord  2)5  =  2  r  sin  a 

=  2r8in  (1^/2); 
Sector  ODCB  =  -^^  irr^,  (a  In 

degrees) ; 
Triangle     I)OB  =  r^  sin  a  cos  o 

=  (l/2)r»sin2o; 
Segment  £>FBC-=  r^  [ira/lSO 

-(sin2a)/21. 
/(I  +cosa)  ; 
—  cos  a)/2 ; 


cos  o)/2. 


ii,  b,  seiniaxes  ;  r,  >•',  radii, 
e  ='\/aJ  -  l^ ; 

(eccentricity); 
p  =  62/,.  =  a(l  -  c')/*- ; 

a  =  tan-»  ('^)  =  eccentric 
angle ; 
a*  =  a  cos  a,  y  =  6  .sin  a  ; 

2,S  ,    .    2.S 

a-  =  a  cos  — p ,  y  =  6  sin  — -  , 

a)  =  a  sin  4",  y  =  6  cos  <<> ; 
ij»  =  w/2  -  a. 


/■  +  /•  =  const.  =  2  a.  .4  =  irrtft  ; 

.S  =  OA />;=—-  a  =  —-  cos  -'  -  ; 
2  2  a ' 


j:v5^ 


rt*  —  a!' 
[  Tablet.] 

=  n  J  Q  v^l  —  f  *  cos'  a  t/o  : 

where  cos  o  =  x/ii. 

rc5/>=PV^^'</^- 

=  a  i     Vl  -#JsinJ<(.  (/<fr 
Jo 

(^  =  rr/2—  a  ;  sin  <^  =  J^/i/. 


10 


STANDARD  FORMULAS 


[II,  F 


Dimensions  or  Equations 


5.  Hyperbola. 

e>l; 


F' 

a 

l^[MF 

N 

1 
/ 

\  n  /   ^ 

6.  Parabola.   e= 


V  P. 


7.  Prism. 


8.  Prismoid 

p.  125). 


a2      &2 
2_ 


(origin  at  O)  (pole  at  F). 


p  =  LN/2 ; 
LiV^=  latus  rectum. 

y^  =  Ipx,  (origin  at  O) ; 
P  =  -. — ^-— s.  (pole  at  F). 


B  =  area  of  base  ; 
A  =  heiglit. 


£  =  lower  base  (area) ; 
M—  middle  section  ; 
7*=  upper  base  ;  /t=  height. 


r-'  —  r  =  const.  =  2  a  ; 
5=Sector  OVP  =  ^  log  (^  +  ^) 

=¥"••'-©-¥•'■■'-■(1)^ 

a!=a.cosh?^,  y  =  &8inh— ; 
ab  (lb 

or  if  tan  <^  =  sinh  ^^ , 

ab 
X  =  asee(j>,  y  =  b  tan  <^. 


Area  ONPM=  § ^2  a;3'2pi/j ; 
Arc  OP  = /"^  Vl  +{:y/p)idy. 
(See  Tables,  p.  3S,  No.  45  (a). 


(See  also  Taft/es,  IV,  G,  p.  A6.) 


[The  volume  of  each  of  the  solids  mentioned  below,  except  (16),  follows  this  formula, 
though  not  all  are  prismoids.] 


9.  Pyramid     (any 
sort). 


10.  Bight  Circular 
Cylinder. 


A  =  area  of  base ; 
h  =  height. 


r  =  radius  of  base  ; 

h  =  height ;  ^=base  (area). 


r=A./>/ii. 


A  (curved)  =  2  nj-h  ; 

A  (total)  ,=  2  nrh  +  2  nr^  ; 


II,  F] 


MENSURATION 

Dimensions  or  EiirATioNS 


11 


11.  Right  Circular 
Cone.  Si'c  Kig.  2 1 ,  p.  s7. 

Urn  a.=r/li  ; 
cos  a=/i/s  ;  .sin  a  =  t/x. 


12.  Frustum    of 
Cone. 

JJ=lower  base  (area); 
7'=iiiii)t'r  base. 


13.  Sphere. 

(a)  Entire  Sphere. 


{b)  Spherical  Seg- 
ment. 

Other  notations  as 
above. 

(c)  Spherical  Zone. 


r  =  radius  of  base  ; 
A  =  height ;  B  =  base ; 
«  =  slant  height ; 
a  =  half  vertex  angle. 


r  =  radius  lower  base  ; 
li  =  radius  upper  base  ; 
/(  =  height;  «=slant  height. 


+  (5  -  ?„)»  =  ,J  ; 
r  =  radius  ;  d  =  diameter  ; 
C  =  great  circle  (area). 


a  =  radius   of  base   of  seg- 
ment ; 
h  =  height  of  segment. 


h  =  height  of  zone  ; 
«,  7)  =  radii  of  base 


A  (curved)=  TrrV/J  +  As 
^  (total)  =7rr(«-f  /■)  ; 
V=-nrVi/Z  =  Bh/i. 


A.  (curved)  =it»(R  +  r)\ 
V=7rh(^Ifl  +  llr  +  r*)/S. 


^  =  4  nri  =  7r(/2  =  4  C; 
>^  =  47rH/.}  =  7r(/3/6 
=  ^  .  r/S  =  4  0/3. 


(j2  =  /( (2  r  -  A) ; 
A  =  2  ttM  =  77  («2  +  /*2)  ; 
F=  ffA.  (3  rt2  +  A2)/6 
=  nh^  (3  r  -  A)/3. 


^  =  2  ttM  ; 
V  =  nh  (S  </J  ■ 


(d)  Spherical  Lune 


=  angle  of  lune  (degrees^. 


^  =  7r7-2a/90. 


(e)  Spherical     Tri 
angle 

Sides  a,  p,  y. 
Angles  A,£,  C. 


E=  A  +  8  +  0-  ISO"; 
S={A  +  B+C)/i\ 
«  =  (a  +  j3  +  y)/2. 


X-  =  V  [sin  (s  -  a)  sin  (s  -  $)  sin  (*  -  7)l/sin  ; 


.r  =  V  -  cos  i/[cos  {S  -  A)  cos  (6'  • 


1(^-0]. 


A  =  nri£:/\SO, 
sin  .4  _  sin  B  _  sin  C. 
sin  a       sin  0       sin  y  ' 
cos  a  =  cos  P  cos  y 

-f  sin  P  sin  y  cos^  ; 
cos  A  —  —  cos  ^  cos  C 

+  sin  £  sin  Ccosa; 
tan  (^4/2)  =  X-/sin  (s  -  a)  ; 
tan  (a/2)  =  A' cos  (.9  -  /I). 


14.  EUipsoid. 

Semiaxes,  a,  I/,  c. 


15.  Paraboloid   of 
Revolution. 

.r2+.'/2  =  2;/2. 


16.  Anchor  King. 


a.y„!  +  ^2/^,2  +  22/^.2=1. 


r=  477(1  ?yf/3. 


r  =  radius  of  base  ; 
h  =  height. 


rVi/2  =77 /./(». 


/•  =  radius,   generating  cir- 
cle; 
B  =  mean  radius  of  ring. 


A  =4  iriJir ; 
F=2  7r«/i/-». 


[See  also  Standard  Applications  of  Integration.,  Tables.,  IV,  H,  p.  46.'\ 


12 


STANDARD  FORMULAS 


[II,  G 


G.   Trigonometric  Relations.     For  Trigonometric  Mensuration  For- 
mulas, see  II,  F,  1,  3,  13  e,  p.  9. 

1.  Definitions.     See  also  II,  F,  3,  p.  9. 

sin  A  =  y/r  ;  cos  A  =  x/r ;  tan  A  =  y/x  ; 

CSC  A  =  r/y  ;  sec  A  =  rjx  ;  ctu  A  =  r/y  ; 

vers  A  =  1  —  cos  A;  exsec  J.  =  sec  ^  —  1. 

2.  Special  Values,  Sigyis,  etc,  for  siiie,  cosine,  and  tangent. 


Angle 

0° 

30^ 

45° 

60° 

90° 

1S0° 

270° 

360°  d=  A 
or  0°±  J 

90°  ±  A 

180°  ±  A 

270°  ±  A 

Bin 

±0 

1/2 

Vi/-2 

V3/2 

1 

±0 

-1 

±sin^ 

+  cos^ 

T  sin  A 

-  cos  A 

cos 

1 

Va/2 

Vi/2 

1/2 

±0 

-1 

±0 

+  cos^ 

T  sin  A 

—  cos  A 

±sin^ 

tan 

±0 

V3/3 

1 

\/3 

±oc 

±0 

±« 

±  tan  A 

=FctnYl 

±  tan  A 

=F  ctn  A 

15. 


0  and  ±  00  indicate  that  the  function  changes  sign.] 

CSC  A  =  1/sin  A;  sec  ^  =  1/cos  A;  tan  ^  =  1/ctn  A. 

x2+?/2  =  »'2:  cos2^+sin2^  =  l;  l+tan'''^=sec2^;ctn2J.  +  l=zcsc2 J.. 

sin  (A  ±  B)  =  sin  A  cos  B  ±  cos  A  sin  B. 

cos  (A±  B)=:  cos  ^  cos  ^  T  sin  A  sin  J5. 

tan  (^  ±  jB)  =  [tan  A  ±  tan  -B]  ^  [1  T  tan  ^  tan  J5]. 

sin  2  ^  =  2  sin  A  cos  ^  ;  sin  a  =  2  sin  (a/2)  cos  («/2). 

cos  2  ^  =  cds2  A  —  sin2  A  =  l-2  sin2  ^  =  2  cos^  J.  —  1 ; 
cos  a  =  cos2  (a/2)  -  sin2  (a/2)  ;  see  also  II,  F,  3,  p.  9. 

sin  3  ^  =  3  sin  ^  -  4  sin^  A.        11.    cos  3  ^  =  4  cos^  ^  —  3  cos  A. 

tan  2  A  =  2  tan  ^  -4-  [1  -  tan2  A].     [See  also  II,  F,  3,  p.  5]. 

2  sin  Acos  B  =  sin  (A  +  B)  +  sin  ( A  -  5)  ; 

sin  a  ±  sin  /3  =  2  sin  [(«  ±  /3)/2]  cos  [(«  T  /3)/2]. 

2  cos  AcosB  =  cos  (A  -  5)  +  cos  (A  +  5)  ; 
cos«  +  cosi3  =  2cos[(rt  +  i3)/2]  cos[(«-  /3)/2]. 

2  sin  A  sin  B  =  cos  (A-  B)-  cos  (J  +  ^)  ; 

cos « -  cos /3  =  -  2sin[(a+/3)/2]  sin  [(a- 3)/2]. 


II,  H] 


TRIGONOMETRY 


13 


16.  sin2  A  —  sin2  B  =  cos-  B  —  cos^  A  =  sin  (A  +  B)  sin  {A  —  B). 

17.  cos^  A  -  sin'^  B  =  cos^  B  —  sin^  A  =  cos  (A  +  B)  cos  (^1  -  B). 
18o   Definitions  of  Inverse  Trigonometric  Functions : 

(o)  y  =  sin-i  X  =  arc   sin   x  =  angle   whose   sine   is   x,    if  x  =  sin  y  ; 
usually  y  is  selected  in  1st  or  4th  quadrant]. 
{b)  y  =  cos-i  X  =  arc  cos  x,  if  x  =  cos  y ;  [take  y  in  1st  or  2d  quadrant], 
(c)  y  =  tan-i  x  =  arc  tan  x,  if  x  =  tan  y ;  [take  y  in  1st  or  4th  quadrant]. 


19.  sin-i  X  =  ir/2  —  cos-i  x  =  cos-i  Vl  -  x-  -  tan-i  [x/Vl  -x-] 

=  csc-i  (l/.r)  =  sec-i[l/\/l  -  .r-]  =  ctn-i[v'l  -  x-'/x]. 

20.  cos-i  X  =  ■7r/2  —  sin-i  x  =  sin-i  Vl  —  x-  -  tan-i  [  Vl  -  x-fxl 


=  sec-i  (1/x)  =  csc-i  [1/Vl  -  x2]  =  ctn-i  [x/ Vl  -  x^}. 

21.  tan-ix  =  7r/2  -ctn-ix  =  ctn-i  (1/x)  =  sin-i  [x/VrTx"^] 

=  cos-i  [l/Vr+T^]  =  sec-iVl  +  x2=:csc-i  [Vl  +x-/x]. 

22.  Special  values,  correct  quadrants,  etc.,  for  inverse  functions. 


Vaute 

+ 

- 

0 

1 

-1 

1/2 

V2/2 

V8/2 

V3/3 

>1 

-fc 

sin-lar 

lst<2 

•1th  Q 

0 

t/2 

-7r/2 

,r/6 

7r/4 

n/S 

0.G2 



-sin-U+X-) 

cos-la- 

UtQ 

2dQ 

n/2 

0 

TT 

7r/3 

ir/4 

n/6 

0.96 



7r-COS-l(+X-) 

tan-l.r 

\»tQ 

4th  (? 

«o 

ttA 

-7r/4 

0.4fi 

0.62 

0.71 

,r/6 

>Tr/4 

-tan-l(+^-) 

H.   Hyperbolic  Functions. 

1.  Definitions.     (See  figures  III,  E,  Jo,  pp.  20,  28  ;  and  V,  C, 
Binh  X  =  (e*  —  e-'^)/2  ;  cosh  x  =  (e*  +  c-')/2 ; 

tanh  X  =  sinh  x/cosh  x  =  (e*  —  e~*)/(e*  +  e**)  ; 
ctnh  X  =  1/tanh  x  ;  sech  x  =  1/cosh  x  ;  csch  x  =  1/sinh  x. 
0  =  Gudermannian  of  x  =  grdx  =  tan-i  (sinh  x)  ;  tan  <p  =  sinh  x. 
=  tan-»[(e'-e-')/2]  =2tan-ie' 

2.  cosh2  X  —  sinh2x  =  l.  3.    1  —  tanh» x  =  sech^ x . 

4.  1  —  ctnh^x  =  csch^x. 

5.  sinh  (x  ±y)  =  sinh  x  cosh  y  ±  cosh  x  sinh  y. 


p.  5f. 


r/2 


14  STANDARD  FORMULAS  [II,  H 

6.  cosh  (x  ±  «/)  =  cosh  x  cosh  y  ±  sinh  x  sinh  y. 

7.  y  =  sinh-ix  =  arg  sinh  x  =  inverse  hyperbolic  sine,  if  x  =  sinh  y. 
[Similar  inverse  forms  corresponding  to  cosh  x,  tanh  x,  etc.] 


8.  sinli-ix  =  cosh-i  vx^  +  1  =  csch -i  (1/x)  =  log  (x  +  Vx'^  +  1). 

9.  cosh-ix  =  sinh-i  Vx^  —  1  =  sech-i  (l/x)  =  log  (x  +Vx^  —  1). 

10.  tanh-ix  =  ctnh-i  (1/x)  =  (1/2)  log  [(1  +  x)/(l  -  x)]. 

11.  If  (j>  =  gd  X,  sinh  x  =  tan  (^,  cosh  x  =  ctn  0,  tanh  x  =  sin  0. 

I.  Analytic  Gecmetry 

[(x,  y)  or  (a,  h)  denote  a  point ;  {a-^,  y{)  and  (3-2,  ^2)  two  points  ;  etc.] 


1.  Distance  I  =  P1P2  =  \/(x2  -  xi)2  +  (^2  -  yiY  =  Vax^  +  a/. 

2.  Projection  of  P1P2  on  Ox  =  Ax  =  X2  —  xi  =  i  cos  a,  where 

«=Z(Or,  P1P2). 

3.  Projection  of  P1P2  on  Oy  =  Ay  =  yo  —  yi  =  I  sin  a. 

4.  Slope  of  P1P2  =  tan  a=  {y^  —  2/i)/(x2  —  Xi)  =  A?// Ax. 

5.  Division  point  of  P1P2  in  ratio  r :    (xi  +  r  Ax,  yi  +  r  Ay). 

6.  Equation  Ax  +  By  +  C  =  0:    straight  line, 
(rf)  y  =  mx  +  b  :   slope,  in  ;  y-intercept,  b. 

(6)  y  —  yo  =  m  (x  —  Xo):   slope,  }?i ;  passes  through  (xq,  j/o)- 
(c)   (y-2/i)/(2/2-2/i)=(x-Xi)/(x2-Xi):  passes  through  (xi,?/i),  (x2,?/2). 
(c?)  X  COS  a  +  y  cos  ^  =p:     distance    to   origin,   p  ;     a  —  Z(Ox,   n)  ; 
/3  =  '^{Oy^  n);  n  =  normal  thi-ough  origin. 

[General  equation  Ax  +  By  +  C  =  0  reduces  to  this  on  division  by  \^A^  +  &.] 

7.  Anglebetweenlinesof slopesTOi, TO2=tan-i[(TOi  — m2)/(l  +  mim2)]. 
[Parallel,  if  TOi  =  m2  ;  perpendicular,  if  l+TOim2=0,  i.e.  if  mi  =  — l/mo.] 

8.  Transformation  x  =  x'  +  h,  y  =  y'  +  k.     [Translation  to  (/;,  k).'] 

9.  Transformation  x  =  ex',  y  =  ky'.     [Increase  of  scale  in  ratio  c  on 
X-axis  ;  in  ratio  k  on  y-axis.  ] 

10.  Transformation,      x  =  x'  cos  ^  —  y'  sin  tf,      y  =  x'  sin  d  +  y'  cos  6. 
[Rotation  of  axes  through  angle  6.'] 

11.  Transfoi-mation  to  polar  coordinates  (p,  d):  x  =  p  cos  0.,  y  —  p  sin  $. 
Reverse  transformation  :   p  =  Vx-^  +  y^,  6  =  tan-i  (y/x). 


II,  J]  ANALYTIC  GEOMETRY  15 

12.  Circle :  (x  —  a)'^  +  (2/  —  b)'^  =  r- ;  center,  (a,  6)  ;  radius,  r ;  or 
{x  —  a)  =  r  cos  e,  {y  —  b)  =  r  sin  ^.     (d  variable.) 

13.  Parabola  :   y'^  =  2px:   vertex  at  origin  ;  latus  rectum  2  p. 

14.  Ellipse :  x-/a^  +  2/-/^"^  =  1  =  center  at  origin  ;  seniia.xes,  a,  b. 
(See  II,  F,  4,  p.  9.) 

15.  Hyperbola :  x^/a^  —  y'^/b'^  =  1  ;  center  at  origin  ;  semiaxes,  a,  b  ; 
asymptotes,  x/a  ±  y/b  =  0.     See  II,  F,  5,  p.  10. 

(a)  If  a  =  ft,  X-  —  y-  =  a" ;  retangular  hyperbola. 
(6)  xy  =  k;  rectangular  hyperbola  ;  asymptotes  :  the  axes, 
(c)  2/  =  (a.»:  +  6)/(ca;+rf),  rectangular  hyperbola;  asymptotes:  x=  ~d/c, 
y  =  a/c. 

16.  Parabolic  Curves:   y  =  ao  +  aix  +  a^x^  +  •••  +  anX". 
[Graph  of  polynomial ;  see  also  Figs.  A,  B,  pp.  17,  IS.} 

17.  Lagrange  Interpolation  Formula.  Given  y  —  f(or),  the  poly- 
nomial approximation  of  degree  »  —  1  [parabolic  curve  through  n  points, 
(a:i,  yi),  (x-2,  yo),  •-,  (a-„,  «/„)]  is 

y  =  P(X)  =  UiPiix)  +  ViP>>{3C)  +  -  +  ynPni.^), 

where  the  polynomials pi(x),  p^{x),  •••,  p«(x)  are 

Pi(x)  =    (a^  -  a:i)(a;  -  xo)  —  (x  -  x.-i)(x  -  Xj+j)  ...(x-Xn) 

(Xf  -  Xi)  (X,-  -  Xo)  ...  (X,-  -  X,_i)  (Xi  -  Xf+i)  ••  •  (Xj  -  x„) 

[Numerator  skips  (x  —  x,)  ;  denominator  skips  (Xj  — x,).  Proof  by 
direct  check.] 

[For  a  variety  of  other  curves,  see  Tables,  III,  pp.  17-32.} 
Formulas  of  Solid  Geometry,  §§  102-3 ;  pp.  31.5-10  ;  see  also  Figs. 
"^III,  N1-X5,  p.  31.     When  possible  the  preceding  formulas  of  plane  geom- 
etry are  so  phrased  that  an  additional  term  of  the  kind  indicated  gives  the 
analogous  formula  of  solid  geometry.     In  particular,  see  G  d,  p.  I4. 

J.     Differential  Formulas. 

[See  (a)  List  nf  Dijfi'rential  Formulas  of  Elementary  Functions,  pp. 
40,  173. 

(6)  List  of  Standard  Integrals,  p.  174,  and  Tables,  IV,  p.  33.  Reverse 
these  to  obtain  Differential  formulas. 

(c)  List  of  Standard  Applications  of  Integration,  Tables.  IV,  H,  p.  46. 

(d)  Infinite  Series,  Taylor's  Formula,  etc.,  see  Tables,  II,  E,  pp.  7-8. 


16  STANDARD  FORMULAS  [II,  J 

1.  2/  =  /(a?)  :    dy  =f'(x)  dx,  f'(x)  =  dy  -^  dx  =  dy  dx. 

2.  F{x,  y)=0:  F^dx  +  Fydy  =  0,  or  dy=-  IF^  ^  Fy]  dx ; 

F^=dF/dx,  F,=  cF/dy. 

3.  35  =/(«),    y  =  <Pit): 

(a)  dx  -f'{t)  dt,    dy  =  <p'(t)  dt,    dy/dx  -  ./.'(«)  -f- /'(«)• 

(6)  d^y/dx:^  =  didy/dxydx  =  (7[<^'  ^f<ydx  =  [0"/'  -/"0']  -  {fy. 

(c)  #2/AZa;3  =  (2[dV<^a;2]/dx  =  di{4>"  f  -f"<t>')  -  (/')']A^<  -/'• 

4.  Transformation  ac  =  /(<) :   y  =  ^(x)  becomes  y  =  <p{f{t))  =  i^  (<). 
(a)  dy/dx  becomes  d2//(7«  -^f'{t)  ;    [see  3  (a)]. 

(&)  dV<^x2    becomes     [(d^f^^')  •/'(«)  -  (.dy/dt)f"(t)}  -^  [/'(«)]'; 
[see  3  (6)]. 

5.  Transformation  ac—f(t,u),  y  =  <j>(t,  u) :    y  =  F(x)   becomes 
u  =  ^{t). 

(a)  dy/dx  becomes  ^^^U  1^  or  \^±  +  ^-±.  ^I  -^  r^+ §^  .  ^]. 
^     ^  di      d<         La«      du     dtA      \_dt     du     dt\ 

(b)  d^y/d£'^  becomes  d  [dy/dx]/dt  -i-  dx/dt ;   [compute  as  in  5  (a)]. 

6.  Polar  Transformation  x  =  pcose,  y  =  p sin 0  . 

dx  =  cos  0  dp  —  p  sm  0d0  ;     dy  —  sin  0  dp  +  p  cos  0  dO, 
d^x  =  cos  0d^p-2  sin  0  dp  dO  -pcosO  d0^, 
d'^y  =  sin  0  d'^p  +  2  cos  ^  dp  dd  -  p  sin  0  dff-. 

1.    z  =  F(x,  y):    dz  =  F^  dx  +  F,  dy  =  p  dx  +  qdy;    [see  I,   3  (d), 
'p.  ^]. 

8.     Transformation  x  =f(u,  u),  y  —  <p(u,  v)  :  z  =  F(x,  y)  —  $(m,  v). 

(  \^_^^,^zd^       dz__dz^  df     ds  d^ 
du~  dxdu     cycu      dv~  dxdv     dy  dv  ' 

(b)  ^  =  A^  +  JB^,     ^=C^+D^. 
dx  cii  dv       dy  du  dv 

[^,  B,  C,  D  found  by  solving  8  (a)  for  dz/dx  and  dz/dyj] 
^  '  aa;2      dx\dxj      dx\     du  dvl 


=  A^A'1-  +  B^^\  +  B^-{a^^  +  B^A' 
du  \      du  cvj  ci>  \     du  dv/ 

[Similar  expressions  for  c-z/dy'^  and  higher  derivatives.] 


TABLE   III 
STANDARD    CURVES 

A.    Curves  y  =  xn,  all  pass  through  (1, 1) ;  positive  powers  also  through 
(0,  0);  negative  powers  asymptotic  to  the  j/-axis.     Special  cases  :  ji  =  0,  1 


Chart  of  ?/  =  x"    for  positive,  negative, 

fractional 

values  of  n 

1          1    '    1 

\\    w 

' 

1  /'   / 

1    1/ 

^z 

1              1 

1 11 J  '\[\ 

.,    X  2 

/!  /I 

I  / 

T-. 

1  1           '  ! 

V,  T  , 

2    -'/_t/j 

■-'/I 

.x7 

'  /'             1 

\    '     1  \  ^1 

1    ii  i" 

n    iimH 

yi  './ 1  1  1  1  1  1 

1              ' 

\>    -\  1 

4i-: 

~   "/^/  ' 

/ 

1  i  1 

\  1       J  \  11 

u 

/_/ 

7  7 

"^             7 

'     '      / 

1  \'*  i  \\~ 

V  t  t 

'I  I/I 

7 

i               '1/ 

\\  \-  '   - \ 

A  V  T 

J 

1 

' 

:'/ 

L      Ij.i  ll. 

111    \    1    1  \ 

TUP 

/ 

/ 

-        ^-^fl 

\  V '  K,r  ■ 

>   il    \  '     1    ' 

n\  \ 

I 

"  ^T^ 

'    \\  j  y     -i 

■sW 

/ 

' 

/    !  .  . 

zW'\iv\\\\ 

/ /i    1/ 

X-  'U^^ 

\  p '~  ~7i 

\V^  AW 

/■' 

\  y^ 

"li   1/ 

_  AvA  ,'\\\\\ 

/ 

1  p^  \ 

t^X     li           /                  1 

/  '^1  JV-.U-1 

\    \        1   / 

~i  \\^ 

^-^ 

\>^ 

\^  A            /         Evt  1  Fo»er9 

U\  -^N 

-rn  ^/  M 

,    >  .   \\  1  /        r"-  I'T'"' "', 

v.// 

,^ 

^^^^ 

^^<i;       \Ml/             2Dd  liuadraiit 

^ 

-~<Wn   =  0               1 

.  ii^JL 

/7  wl^S^' 

"T^ 

r/ 

1^^>/ 

^^=4l3i: 

^v'<t- 

ii^i^M^^^    - 

Kv^^ 

^^'/w,     ^^^~~4^ 

y^i  -/i     \\'  1   ^. 

/7'W' 

\\\K7 

r~--il-''"     ..          II 

'  //  y  \  wv,    \ 

i^TV 

y/y/i 

\\\  ^ 

-"j"""^^^ 

Y/A. 

^v/ / 

i~^^ 

^f-i:i^ 

^3^^^-^ 

'i/M 

2:^^^^^^^ 

■  i^~-i 

— r-^    • — t— 

1 1   1  1 1 1 1 1   1^^  > 

■"^!     1     1 

{{111 

1   1 

1    i    1    i                ^ 

OdJ  I'owen  all.i  Iluots      -^      j       / 

yi^b^ 

1        ",01 

"^  U- 

«X-U^ 

1   r...ma,.i„|      1   ^            ^/ 

\  1"' 

^•> 

4— :i= 

b^^fe^ 

\      ]3rd  C  u  Jnihl    /,  ■•        './ 

'   v< 

t'^ 

*tii^- 

\[       i  [A    y     . 

\sx 

li 

V^l     "; 

1^'" 

'  k 

%     M    ^     "^" 

\*lS 

^ 

Vi-     n  = 

1.11, 

- 

Adi.baU^  EkpA,.lon  for  Air 

-^^^  -f-  ^  -^y 

NtT 

^z- 

j*-K'  1 

1 

M  ''^^'A^ 

's£snn 

^i\' 

1 

liX^^t^ 

n  '\.  fv^  \  p/\    '\Jy 

"^ 

fT= 

— ir^:^ 

\~7^^^^IA^^^  \ 

1  //; 

3t_d 

^"^^^^VT^ 

— >-f--' 

yf'C^ 

cSi  T^ 

sT 

' -1-— 

4>'4/vf^  ^. 

'^^  ^^ 

7 

\ 

"^ 

^~-^'"', )          "' 

ii  itifi=§v    \    " 

'^  V 

7 

>\* 

^^^\^ 

7         \\^\       \     ~ 

"  ^  i 

1% 

1        I               1        ^^-<3       ■/    i 

4  \\  '^     A  , 

\M  ^^ 

J 

V 

i 

i 

1       '"i^ 

it       "        AH- 

^w  ;? 

^ve  1  ltou(>. 

i 

1 

4it  +-    t    ^^ 

t  if  V.' 

r  laiipear  lu 

1 

1 

n:  "   iH    '%■' 

•g_5 

Ih  QuDdriin 

1    1    \\\ 

1 

1 

\\         \\      '\ 

/ 

-t-r 

1  1  l'\ 

1 

_L 

Fia.  A 
17 


18 


STANDARD  CURVES 


[III,  A 


are  straight  lines  ;  n  =  2,  1/2  are  ordinary  parabolas  ;  re  =  —  1  is  an 
ordinary  hyperbola  ;  n  =  3/2,  2/3  are  semi-cubical  parabolas. 

The  curves  pv^  =  c  occur  in  the  theory  of  g-as  expansion,  where  /^  =  pressure; 
•»  =  volume  ;  c  and  m  constants.  In  isothermal  expansion  (p.  29'2)  jn  =  1,  whence  pv  =  c 
orp  =  c»— 1;  (?i  =  —  1  in  Fig.  A).     Choose  scales  so  that  y  =  ^Vc  and  «  =  a-.    Inadiabatia 


':■  done 


compressing 


J"  "2 
j}dc  =  worA- 

B.  Logarithmic  Paper  ;  Curves  y  =  .r",  y  =  Jcx**^.  Logarithmic 
paper  is  used  chiefly  in  experimental  determination  of  the  constants  k  and 
n  ;  and  for  graphical  tables.     In  Fig.  B,  k  =  1  except  where  given. 


y 

q^B 

iMimwt 

\ 

\ 

1    i^ 

M/!!l/    1/   V 

■ 

/' 

8       ^ 

>kl,l  1  i\\v. 

^ 

o    M 

r  f  1 

/ 

/ 

.°       '^^> 

:]TissiA\\v 

V 

W 

fi 

/ 

/ 

/ 

^^MJiApvY^ 

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r  thm  c  Chart  ^ 

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^ 

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nl   Pjfe^*" 

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1  j\  i  \ 

\  \  ^«^^i\ 

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■^ 

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A  / 

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f^ 

il 

^v 

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n  :\- 

l  \J  \-?  Ni 

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1 

v.. 

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i| 

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ttlt-? 

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V 

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y  /    171  1 

I\^llu 

1  1  ■! 

i:::HZ 

^ 

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>,      1  i-<^i^ 

•la      .2    .26  :i 


.1.5         2t      2.5     3   3.5  4  4.5  5      6     7    8    9  1» 


Fig.  B 


{See  §122,  p. 229,  above;  also  Williams-Hazeu,  ffydraulic  Tahlex ;  Trautwine,  En- 
gineers^  Handbook  ;  D'Ocagne,  Nomographie.\  The  line  y  =  a*-— 1  gives  the  reciprocals 
of  numbers  by  direct  readings. 


Ill,  D] 


ELEMENTARY  FUNCTIONS 


19 


C.    Trigonometric  Functions.     The  inverse  trigonometric   func- 
tions are  given  by  reading  y  first. 


:::+q: 

1   ! 

tF 

EEEir 

m 

N'^ 

^ 

w 

S'  - 

#1= 

* 

|±W^ 

1 

jt 

^|: 

Ff* 

Iv^f/ 

T 

i 

V 

^ 

f" 

1  ^    ;■ 

^■4- 

<- 

^^ 

^ 

''^ 

y 

^ 

7  "i'   ■"  • 

7^'" 

-  ^- \'>'^ 

1 

y- 

.\ 

A 

/ 

'if 

-:Hch. 

Trofthe  '      L 

trin   Fnnrlir^na 

r^ 

^ 

(>; 

and  Their  Inverses 
8  y  =-sm  .r    gives  x=  arc 

_ 

r\ 

y^ 

vX~^ 

'A -- .. 

I 

Z  /-/ 

\ 

^-^- 

-f^^+7-f 

m  !/  L 

\ 1 

M   1   ! 

1    ;^       '          K      - 

' 

' 

1  /< 

\     :     ,          I 

Fio.  C 

D.    Logarithms  and  Exponentials  :  y  —  logiox  and  y  =  log, a;. 
Note  logeX  =  logioX  logg  10  =  2.303  logio  .r..     The  vahies  of  the  expo- 
nential functions  x  =  10*'  and  x  =  e"  are  given  by  reading  y  first.     See  E. 


20 


STANDARD  CURVES 


[III,  E 


E.    Exponential    and    Hyperbolic     Functions.       The    catenary 

(hyperbolic  cosine)  [?/ =  cosh  x  =  (e^  +  e-^)/2J  and  the  hyperbolic 
sine  [y  =  sinh  x  =  (e^— e-^)/2]  are  shown  in  their  relation  to  the  ex- 
ponential curves  y  =e'',  y  =  e"*.  Notice  that  both  hyperbolic  curves 
are  asymptotic  to  »/  =  e^/2. 


Fig.  E 


The  curve  y  =  e~'  is  the  standard  damping  curve  ;  see  Fig.  F2,  and  §  92, 
p.  160. 

The  general  catenary  is  y  ={a/'i,){e'/'^  +  e~*/")=  a  cosh  (ar/a)  ;  it  is  the  curve  in 
which  a  flexible  inelastic  cord  will  hang.     (Change  the  scale  from  1  to  a  on  both  axes.) 


Ill,  F] 


HARMONIC  CURVES 


21 


F.    Harmonic  Curves.     The  general  type  of  simple  harmonic  curve 

is  2/  =  a  sin  (A-.r  +  e): 


Curve 

[  a  sin  (&x  +  «) 

sinx 

cosx 

8in2jp 

(1/2)  sin  (6* - 

- 1.2; 

aiiiplitiide 

a 

1 

1 

1 

1/2 

wave-leniarth 

!            2  ,r/^- 

•2  IT 

2. 

jr 

./3 

phase 

-  f/'!- 

0 

f/'i 

0 

0.2 

A  compound  harmonic  curve  is  formed  by  superposing  simple  har- 
monics :  in  Fig.  Fi,  j/  =  sin  2  x  +  (1/2)  sin  (6  x  —  1.2)  is  drawn. 


-Jl\- 

y-z 

=  Bini2x 
=  4-  8iu(0r- 

1.2) 

i 

r 

'V 

=    8in2z+l.  8 

1 

^    1 

n  (Cx- 

i 

/ 

\ 

A". 

\^ 

^h\ 

/v. 

h 

\\. 

^'A 

■\/ 

V 

K 

w 

/l/i 

1 

^ 

V 

!        1        ! 

Such  curves  occur  In  theories  of  vibrations,  sound,  electricity.    See  §§  90-92,  pp.  157 


22 


STANDARD  CURVES 


[III,  F 


The  simplest  type  of  damped  vibrations  is  y  =  e-"  sin  kx :  Fig.  F2 
shows  y  —  e-^''^  sin  3  x.  The  general  form  is  ?/  =  aer""^  sin  (/tx  +  e) .  Such 
damped  simple  vibrations  may  be  superposed  on  other  damped  or  un- 
damped vibrations.     See  §§  92,  189,  pp.  160,  368. 


- 

4kr 

„l„  ' 

v=JJJ-, 

n 

" 

- 

■ 

IPL 

^ 

i-iauo 

1 

^  ii 

V 

?/  = 

L 

« 

linSa 

> 

\ 

• 

< 

' 

<•>; 

, 

- 

v 

, 

' 

'H 

-•- 

e^ 

- 

I* 

,' 

«>^ 

. 

/ 

s  \T 

. 

' 

V 

, 

. 

, 

' 

_ 

_ 

y 

_ 

I 

_ 

^ 

_ 

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^ 

^_ 

, 

_ 

_ 

_ 

_ 

^ 

pi 

L 

^ 

^ 

_ 

_ 

L 

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_^ 

_ 

_ 

- 

- 

- 

- 

/ 

- 

- 

^ 

- 

- 

- 

1^- 

- 

- 

- 

- 

- 

- 

- 

[^ 

- 

^ 

^ 

'^ 

-^ 

- 

4^ 

^H 

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!- 

- 

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^ 

V 

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s 

s. 

, 

^t^ 

^ 

" 

-■ 

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2j 

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^ 

^ 

^ 

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- 

"1 

- 

A 

- 

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^- 

Kt 

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Y/ 

- 

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: 

r 

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^ 

^ 

r 

- 

^ 

- 

n 

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'- 

- 

- 

\ 

' ' 

Si 

t 

' 

A 

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- 

r 

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1 

^1 

/ 

i 

^ 

1 

', 

- 

,' 

« 

/ 

■' 

- 

- 

- 

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.-/ 

\ 

? 

■  - 

. 

^ 

»^> 

•^ 

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\A 

1 

Fig.  F2 


G.   The  Roulettes. 


A  roulette  is  the  path  of  any  point  rigidly  connected  with  a  moving 
curve  which  rolls  without  slipping  on  another  (fixed)  curve. 


J 

1  ^ 

Ihe  Cycloid 

i^'^ ) 

/ 

0 

M    N 

Fig.  Gi 


Figure  Gi  shows  the  ordinary  cycloid,  a  roulette  formed  by  a  point 
P  on  the  rim  of  a  wheel  of  radius  a,  which  rolls  on  a  straight  line  OX 
See  also  Fig.  69,  p.  .307.     The  equations  are 


[ X  =  ON—  MN  =ae-  asmi 
\ y  =  NC  —  EC  =  a  —  a  cose, 


where  e  =  Z  NOP. 


Ill,  G] 


ROULETTES 


23 


Figure  Go  shows  the  curves  traced  by  a  point  on  a  spoke  of  tlie 
wheel  of  Fig.  II,  or  the  spoke  produced.  These  are  called  trochoids  ; 
their  equations  are 

Ix  =  (10  —  b  sin^, 

\y  =  a—  b  cos  0, 


The  Trochoids 


where  b  is  the  distance  PC-    If  ft  >  a,  the  curve  is  called  an  epitrochoid; 
if  6  <  ff,  a  hypotrochoid. 

Figure  G3  shows  the  epicycloid  ; 

3;=  (a  +  6)costf-/)Cosr^^-±-^(?1, 

y]=  (a  +  ft)  sin  ^  -  6  sin  f^-i-^  0~\, 


Epicycloid 


Fig.  G, 


formed  by  a  point  on  the  circumference  of  a  circle  of  radius  ft  rolling  on 
the  exterior  of  a  circle  of  radius  a. 


24 


STANDARD  CURVES  [III,  G 

Figure  G4  shows  the  special  epicycloid,  a  =  b, 

ix  =  2a cos 0  —  a  cos  2  ff, 
\y  =  2asind  —  a  sin  2 d, 

which  is  called  the  cardioid  ;  its  equation  in 
polar  coordinates  (p,  <p)  with  pole  at  0'  is 
p  =  2a(l  —  cos<^). 


ix  =  (a-b)cose  +  b  cos  [- — -  el, 
\  y  =  (a-b)  sine- b  sin  ^^^^  el. , 

formed  by  a  point  on  the  circumference 
of  a  circle  of  radius  b  rolling  on  the 
interior  of  a  circle  of  radius  a. 

Hypocycloid 


Fig.  Gs 


Figure  Gs  shows  the  special  hypocycloid,  a  =  4b, 
'  X  =  a  cos3  ( 


y  =  a  sm** 


or    x^'«  +  y" 


Fig.  Gs 


which   is  called  the  four-cusped  hypocycloid,  or 
astroid. 


H.  The  Tractrix.  This 
curve  is  the  path  of  a  particle 
P  drawn  by  a  cord  PQ  of  fixed 
length  a  attached  to  a  point 
Q  which  moves  along  the 
3:-axis  from  0  to  ±  00.  Its 
equation  is 


Tbe  Tracttix, 


a:  =  alog"'  +  ^"^^-^'^^/^ 


Ill,  I] 


CUBICS  — CONTOUR  LINES 


25 


I.    Cubic  and  Quartic  Curves. 

Figure  Ii  shows  the  contour  lines  of  the  surface  z=:ji^  —  3x  —  y^  cut 
out  by  the  planes  z  =  k,  for  k=—6,  —4,  —2,  0,  2,  4;  that  is,  the 
cubic  curves  x'  —  3  a:  —  y-  —  k. 


The  surfoce  has  a  maxiinuin  at  a-  =  —  1,  y  =  0  ;  the  point  a"  =  1,  >/  =  0  is  also  a  critical 
point,  but  the  surface  cuts  throu^'h  its  tangent  plane  there,  along  the  curve  A"  =  —  2 ; 
yi  =  a^  _  3  ar  +  2. 

These  curves  are  drawn  by  means  of  the  auxiliary  curve  q=ai'—&x,  itself  a  type  of  cubic 
curve  ;  then  y  =^/q  —  A;  is  readily  computed. 


Coiltpnr  JLines 
of  the'~-^urface 

z-x' 

-3x  + 
z=k 

'\ 

4) 

if  1 

^\ 

< 

1 

\ 

icii 

Jff 

(( 

m 

=  2   (jN 

^ 

n- . 

■^t— 

k^ 

zS-^ 

\    y^' 

<  i 

\K-  = 

^^\\ 

m 

Curv 

s: 

X 

-3x- 

k 

m 

"      1   4-3-1 

t: 

6,-4,-2,  0,  2,  4 
2:    The  Btrophoid 

1 

l.^ 

ixillary  Curves 

-3xaj,d2/l4-x 

1 

1 

Fig.  1. 


Figure  !•>  shows  the  contour  lines  of  the  surface  2  =  :r8— 8  r  +  .v'- (r— 4) 
tor  z  ~  k  =  —  a,  —  4,  -  2,  0,  2,  4  ;  that  is,  the  cubic  curves 


-.(.rs. 


■/0/(4 -.-•). 


The  surface  has  a  maximum  at  (-  1,  0).  At  (1.  0)  the  horizontal  tangent  plane  ?  =  -2 
cuts  the  surface  in  the  strophoid  y*  =  (a^  -  Sai  +  2)/ (4  —  a")  whose  equation  with  the 
r ew  origin  0*  is  jf*  —  a-'^  (3  +  x')  /  (3  -  a;').  The  line  ar  =  4  Is  an  asymptote  for  each  of  the 
curves. 


26 


STANDARD  CURVES 


[HI,  1 


Figure  I3  shows  another  cubic :  the  cissoid, 
famous  for  its  use  in  the  ancient  problem  of 
the  "  duplication  of  the  cube."     Its  equation  is 

or    p  =  2  a  tan  e  sec  6. 


2  a  —  X 


It  can  be  drawn  by  using  an  auxiliary  curve  as  above ; 
or  by  means  of  its  geometric  definition  :  0P=  QB,  when 
Oy  and  AB  are  vertical  tangents  to  the  circle  OQA. 


Figure  I4  shows  the  conchoid  of  Nicomedes,  used  by  the  ancients  in 
the  problem  of  trisection  of  an  angle.     Its  equation  is 


\x-a} 


asec0  ±  b. 


Conchoid 


Folium 

of 
Descartes 


Fig.  I4 


Fig.  Is 


Figure  Ij  shows  the  cubic  x^  +  y^  —  S  axy  =  0,  called  the  Folium 
of  Descartes  J  see  Exs.  1,  p.  46;  11-12,  p.  63. 


Fig.  I. 


Figure  le  shows  the  witch  of  Agnesi :  y  =  8  a^/(x^  +  4  a^)  ;  see  Exs. 

3,  p.  163 ;  5  (ft),  5  (d),  P-  166  ;  5,  p.  180  ;  and  see  III,  J,  below. 


Ill,  I] 


QUARTICS  — CONTOUR  LINES 


27 


Figure  I7  shows  the  Cassinian  ovals,  defined  geometrically  by  the 
equation  PF  •  PF^  =  k- ;  or  by  the  quartic  equation, 

lix  -  ay-  +  2/2]  [(X  +  ay-  +  y^]  =  ^4, 


where  a  =  OF  ( =  1  in  Fig.  I7) .     The  special  oval  k-  =  a^  is  called  the 
lemmiscate,  (x-  +  y-)'^  =  2a^  (x-  —  y-)  or  p-  =  2  a-  cos  2  0. 


\  1 

1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  M  1  1  1  1  1  1  1  1 

Contour  Lin 

es  on  Surface 

2<  = 

-1  (Cx+a)-+^2] 

1 

^     '"  1  1  1  1  1   1  1  1  1  1  1   M   1  1  1  1  1  1 

k- 

•i                       -.         1 

^^ 

--A 

A-  ==p==      -"t="    = 

---'v-^i^'^^" 

X-''           "^^  "^ 

^y               \_. 1— -_^           -^       \^ 

/ 1^}^^  riri 

-^■ 

4J^  ^^^          '^^sN     V- 

J~l     t/L^^^ 

^""r^" 

L  j/           /^^  -J —  -v  "  .  [V     V 

:;   u^^^-(L5^ 

.;>>[                   r      '/}■!■'              \    \    'rt 

-.    AXX-^X- 

/n  1       '^  1  1  r^      \  \\ 

04  FTHiX 

\j    1               T  l?^    i      hV      D   I/I 

-i  -^  ^c^^trzi 

7    -aJZ 

iV?"-''          V  •'  \l  ^  1        \/    1  11 

^          5      S^>4!'J^ 

is.^  '        >_  ^^~>— f-^  ^ /a    I 

^^^^=4-ru 

z>-p^ 

"T^H  1^  '  ^"^"^^^  ■^  ^<  1    y 

V^  ^^sXiii 

I-/  ^■'~>j~ ---v^'     1  ^-r^i  /  y 

-S  ^^  -^=- 

"     I        1 

s  ^-,ir 

_       A;"  = 

1 5                 \^^^  y    . 

15. X— 

_,.„ 

1  ^-' 

1    ,    r  — 

—*   ^   1       ^ 

I'-^l  1  1  1  1  l-H-n  MM 

Cagslnian  Ova 

la  li:x.-a)=- 
(Figure  dra 

-1/-  1  [(X4a)-  +  V"  )  =  A* 

(oc  PF-  PF'-  it-l 

1  M  1  1  1  1 

«„fora=n     1    1    1    1    1    1    1    1 

Fig.  I, 
The  ovals  are  also  the  contour  lines  of  the  surface 

2*=[(x-a)2  +  y2][(x  +  a)2  +  2/2], 


which  has    minima   at    (x  =  ±  a,  y  =  0),    and  a   critical   point   with  no 
extreme  at  origin. 


28 


STANDARD  CURVES 


[III,  J 


V^ 

y 

^v 

JT         /, 

^^                   -jTf 

w^ 

^ 

0 

^1   ^, 

X 

J.   Error  or  Probability  Curves. 

Figure  Ji  is  the  so-called  curve  of  error,  or  probability  curve 

h  ,,2,2 

where  h  is  the  measure 
of  precision.  See  Tables, 
IV,  H,  148,  p.  45;  and  V, 
G,  p.  54. 

Figure  Jo  shows  the  very- 
similar  curve 
Fig.  Ji    ■        " 

y  =  sech  x  =  2/(e^  +  e-^). 

In  some  instances  this  curve,  or  the  witch  (Fig.  le) ,  may  be  used  in  place 
of  Fig.  Ji.  Any  of 
these  curves,  on  a 
proper  scale,  give 
good  approximations 
to  the  probable  dis- 
tribution of  any  ac- 
cidental data  which 
tend  to  group  them- 
selves about  a  mean.  Fiu.  J2 

K.   Polynomial  Approximations. 

Figure  Ki  shows  the  first  Taylor  polynomial  approximations  to  the 

function  y  =  sin  x.     (See  §  134,  p.  2-58.) 


M-]  1  4+/.0-                  U              71  h-L     :      «-,                V    1     ,N\irh 

T     --      -i  --  ^^---     -          -  "fl^, u',r     "            t     Vx^>.   ■" 

A _       : :      'tf-  -  '     -     -   -       -      ^/^                 '  }^~    '^ ' :^   ^ 

li^'l^^'  "  ^  ^*A '' "J^r""  '^^~^^•^   r 

+-f  -      -'A                 -      '                               "        V                ■'T-^L.  ■_p5'<-i2r: 

tc— ^-^iii- L._L--  _j.:::;::±ji±Ei:±± 

Ill,  L] 


APPROXIMATION  CURVES 


29 


Figure  Ko  shows  the  Simpson-Lagrange  approximations  :  (1)  bj 
a  broken  line ;  (2)  by  an  ordinary  parabola ;  (3)  by  a  cubic,  which 
however  degenerates 
into  a  parabola  in  this 
example.  (Lagrange 
Interpolation  For- 
mula, Taft^es,  p.  15.) 

The  fourth  approxi- 
mation is  so  close  that  it 
cannot  be  drawn  in  the 
figure.  In  practice,  the 
division  points  are  taken 
closer  together  than  is  E 
feasible  in  a  figure. 


1  1  1  1  M  1  1  1  1  1  M  M  1  M  i  1  1  1  I  [  1  I  !  i  •  M  rp  n 

Simpson  -  Lagrange  Polynomial  Approximations  \-[y[-\ 

4-      -U- 

-pioi   ij-  s.lixp 

HT 

--i 

±._:^ 

f-r-- 

--4 

+  ----: 

r^d"- 

ntU 

-No.2;r 

.  .^.^.-io-i^ 

■'fc"*! 

II  ^^ 

t:.^n.i  :^[ 

^ 

-i-l<^  .^C 



f4i 

T^  < 

>  ^ 

-y'^Y 

-     ^-^^      A 

t 

-J: 

i     1  1 

P            >C 

\t 

No 

.3      S^        J     J 

.    ::z^_    :il! 

...    ::^    :: 

-S,^ 

^                     Tt 

^     d-       Jtu- 

1^^ 

-7^      -      -          -jf 

--3       --2          - 

^      3 

.^41 

:     ±  iL     :± 

Fig.  Kj 


No.  1.     y  =  -,r,  0<.)-<-; 


y  =  ix  -  A.r2  =  1.2T3a--  AObx^,  Q<x<n. 
(Division  points  •  .t„  =  0,  a-j  =  7r/2,  a-j  =  ir.) 


9%/:?         OV.S 


0-a^  =  1.245ar-  0.895  a-'. 


(Division  points  :  a-g  =  0,  a^i  =  ir/3,  aij  =  2  tt/S,  .Tj  =  tt.) 
No.  4.    y  =  15..38  aj  -  7.642  a'2  -  2.30  a^  -I-  0.3T6  x*. 
(.\grees  too  closely  to  show  in  the  figure.) 

L.    Trigonometric  Approximations. 


Fig.  L 
Figure  L  shows  the  approximation  to  the  two  detached  line-segments 
y  =  —  7r/2,  (—  r  <  a-<  0"),  ?/  =  7r/2,  (0  <  .r  <  w)  by  means  of  an  expres- 
sion of  the  form  Cq  -|-  ai  sin  a;  -|-  aj  sin  2  x  -f  •••  +  a„  sin  tix.    See  II,  E,  26. 


30 


STANDARD  CURVES 


[III,  L 


Coiacideat  Curves 


Fig.  Ml 


Fig.  M2 


Ill,  N]  SPIRALS  — QUADRIC  SURFACES  31 

M.    Spirals. 

Figure  Mi  shows  the  logarithmic  [or  equiangular  spirals  p  =  Af'^] 
for  several  values  of  k  and  a.  Note  that  k  —  e"  and  k=—  \  give  the 
same  curve.     (See  §  96,  p.  108.) 


Fig.  Ma 

Figures  M.2  and  Mg  represent  the  Archimedean  Spiral  p 
Hyperbolic  Spijal  pd  =  a,  respectively. 


?,  and  the 


N.    Quadric  Surfaces. 

These  are  standard  ligures  of  the  usual  equations. 

Hyperboloid  of  one  Sheet 


Ellipsoid                * 

/>'-""/" 

1     a              \x 

\^^^\^,.''& 

__X-</ 

X«            W«            2" 

-^^^ 

Fig.  Ni 


Fig.  N, 


32 


STANDARD  INTEGRALS 


[in,  N 


Hyperboloid  of  two  Sheets 


\ 

Contour 

zy  -■ 

Lines 
z 

P 

\ 

/T"  ■'  II 

2=0 

^ 

^ 

^^\^ 

% 

"'Z-- 

'^^- 

> 

\\\ 

i'/V 

M 

1"    / 

Ii!  / 

Fig.  Na 


Fig.  Ns 


Elliptic  Paraboloid 


Fig.  IS!* 


Fig.  Ns 


TABLE   lY 

STANDARD   INTEGRALS 
Index: 

A.  Fundamental  General  Formulas,  p.  33. 

B.  Integrand  —  Rational  Algebraic,  p.  34. 

C.  Integrand  Irrational,  p.  37. 

(a)  Linear  radical  r  =  Vax  +  h,  p.  37. 

(b)  Quadratic  radical  V±  r-  ±  a^,  p.  37. 

D.  Binomial  Differentials  —  Keduction  Formulas,  p.  39. 

E.  Integrand  Transcendental,  p.  30. 

(a)  Ti-igonometric,  p.  39. 

(b)  Trigonometric  —  Algebraic,  p.  4S. 

(c)  Inverse  Trigonometric,  p.  4.h 

(d)  Exponential  and  Logarithmic,  p.  43. 

F.  Important  Definite  Integrals,  p.  44. 

G.  Approximation  Formulas,  p.  45. 
H.    Standard  Applications,  p.  46. 

A.    Fundamental  General  Formulas. 

1.    If  =  — -,   then  W  =  V  +  constant.       [Fundamental  Theorem.) 


2.  If    I  M  dx  -  1,  then   —  =  U.      [General  Check] 

•^  dx 

3.  \cu  dx  =  c\u  dx. 

4.  (  [u  +  v]  dx  =  \u  d.r  +  \v  dx. 

5.  (  II  dv  =  UV  —  \  V  du.      [Parts] 

6.  {(f(u)du]  =  r/[d)(.r)]  ^'^^-^^  rfX.    [Substitution.] 

LJ  J  u=,uz)     J  dx 


34  STANDARD  INTEGRALS  [IV,  B 

B .   Integrand — Rational  Algebraic . 

7.     fic»»dic  =  ^^,  w^-1,  sees. 
J  n+1 

Notes,  (a)  /  (Any  Polynomial)  dx,  — use  3,  4,  T. 

(6)  /(Product  of  Two  Polynomials)  «?«,— expand,  then  use  3,  4,  7. 
(c)  Jc  rfaj  =  ca;,  by  3,  7. 

8.     C^  =  loge  iC  =  (logio  x)(log,  10)  =  (2.302585)  logio  «. 
J   oc 

Notes,  (a)    J  (l/a-"")  cfx,  —  use  1  with  w  =  —  m  if  ??!  :^  1  ;  use  8  if  m  =  1. 
(P)  /  [(Any  Polynomial)/a''"]  dx,  —  use  short  division,  then  7  and  8. 


9.    r_^=larctan^  =  ltan-i^  =  lctn-i 
J  a'^+3c'^     a  a     a  a     a  ; 


=  —  — Ctn-l— [+ const.]. 

a  a 

10.  r  _^  =  _L  log  ^^:^=-i-  log  ^^:^[+ const.]. 

Note,    All  rational  functions  are  integrated  by  reductions  to  7,  8,  9.    The  reduc- 
tions are  performed  by  3,  4,  6.    No.  10  and  all  that  follow  are  results  of  this  process. 

11.  C(ax  +  ?))»(;x  =  -'^^^  +  ^^"^\n^-l.    (See  No.  12.)  [From  7.] 
J  a      n  +  1 

12.  r — ^ =  1  log  (ax +  6).     [From  8.] 

J  (ax  +  b)      a 

Notes,     (o)   \^^  '^     dx,  —  use  long  division,  then  7  and  12. 
i  ax+  b 

^^^   r  Any  Polynomial  ^^^  _  ^^^  ^^^^  division,  then  7  and  12. 
••  ax  +  b 

13.  r ^?£ =  1 Til („j^l),      [From  11.] 

J  (ax +  6)"'     a  (ni-l)(ax  + &)"'-! 

14.  f_E^?^ —  =  lr_^+log(«x+  &)]•     [Fromll,  12.] 

Notes.     (<()   f  "^''' + -^  tfa-,  —  combine  A  times  No.  14  and  B  times  No.  13,  m  =  2. 
J  (<«•  +  6)2 


(Any  Polynomial) 


J  («»  + 


rfflj,  —  use  long  division,  then  7  and  14  (a);  or  use  15 


IV,  B]  RATIONAL  ALGEBRAIC  35 

15.  [  (  F(x,ax  +  b)dx~\  =  I  C f  (^^-^ ,  u\  du.     [From  6.] 

Notes,    (a)  Restatement :  put  u  for  ai  +  b,  ^^^-^ —  for  a:,—  for  rfce. 

a  a 

(ft)  >/(««•  +  m  d<r.,  -  use  15.   Ane.  ^,  [  "  ^  +  ^J„=a+5.- 

(^,j   [-(Any  Polynomial)^     _         ^5      j^         ^^ 

(^j  |. (Any  Polynomial)  ^^^  _  ^^^  jg  ^^^jg^^  ^  ^3   .  ^^^  ^^^  ^2  (&),  U  (6). 

■•        (aa;  +  h)"^ 
(e)   Jj"  (rta-  +  6)"»  dx,  —  use  15  if  771  >  n ;  use  1  (h)  if  m  <  n  ;  see  also  51-54. 

16.    1 = 1 [— « ^— 1. 

(aa?  +  ft)  (cic  +  d)     ad—  be  Lax  +  b     cx  +  dj 

Notes,    (a)  [ — ,  — use  16,  then  12.        Special  cases,  —  see  10  andl6  (6V 

(ft)   r iLr =  ^_rri (L — Idx.     (Special  case  of  16  (a).) 

(c)  r Ax  +  B rfa5,  —  use  16,  then  long  division,  12. 

^  '  i(ax  +  b)(cx+d) 

(^^   ,(An.vPolvnomi.il)^     ^^^^  jg   ^^^^  ^        division,  7,  12. 
J  Uix  +  b)icx  +ii) 

(e)  If  ad  -  be  =  0,  18  can  be  used. 

Notes.    («)  Restatement : 

PutMfor221+i;  -^fora- 
a;  ?<  —  (i 

(6)  r — — — = — ^— r^"""^" 

•*  .7-"(rtJ-  +  ft)"*  ftm+n-lJ  u"> 

I  ,\   r         dx         u-  a  log  ;<■        .       r 

18.    f     ^^      =  -J—  tan-1  X  Jg ,  if  a  >  0,  6  >  0.    [See  9.] 
J  ax'^  +  b      ^/Wh  ^  b 


-J^[u- 

-a' 

M  - 

-«; 

(u-a)a 

;      *"    for  a 

r  +  6 

(^ 

bdu 

-«)2 

for  daj. 

n+n-2  _, 

du  •  then  U8< 

8  6. 

dx 

«J 

-  if' 

?{  +  2  a«  lopr « 

x^U'x  +  ft) 

2ft3 

log  ^ax-v-b  ^  if  „  >  0,  6  <  0.     [See  10.] 


2  V- a6        Vax+V-b 

dx 


Notes,     (a)  f     °        .  -  use  18  (2nd  part)  ;  ft  =  -  c.     (ft)  f  — ^  -  -  I  -^ . 

J  ««'  -  c  I       '  '  ^   'J  c  -  nx'  J  (7x2  -  c 


36  STANDARD   INTEGRALS  [IV,  B 

J  ax/  +  b      2  a 
Notes,    (a)   [   \      .  ,  —  use  long  division,  then  18. 
(6)   r±liL±L^  ax,  —  use  18,  19. 

(0  J^ 


aa-2  +  6 


OQ     1 1  r      m'^      _     wix  —  ?i~[ 


(WIX  +  7l)(( 


Notes.     («)    f- — — -tt  <? a',  —  use  20,  then  12,  IS,  19. 

^  ■^  (ma!  +  ?0(«a:2  +  i^;      a  7?!a' +  «      m  (!a2  +  i  "^  V  <t        m  )  ()na-+7i)( 

(c)    f- — "y     °  f "°/""^     dx,  —  use  long  division,  then  20  6,  12,  18,  20a. 
^      J  {mx  +  n){ax^  +  ?')  >      >      > 

21.   ax2  +  6x  +  c  =  arx  +  ^]'-^!^il^. 
L        2  a  J  4  a 

Notes,     (a)   f  f       ,   '^f     ,     1  ^  =  f ^^^^^ — ; —  ,  then  18. 

^       J      «a'2  +  b;e  4-  c  I  ,     ,  ,    ^       J        „      b'  -  4<ie' 

(6)    fj^Ca^,   rwJ +  ?;«•  + c)rfa-1    _         ^   =  J /-(„  _  1- ,  „„2  _  ?!izJjI£^  rf„, 

(c)  J 


Xacc'+fi) 


Any  Polynomial 


(<^)  J^^ — r — _  '      •(?»,— long  division,  then  find  one  real  factor  of  cubic,  then  use 

21,  21  f>,     [If  the  cubic  has  a  double  factor,  set  ti  =  that  factor,  then  use  17  c] 


22     f     ^^^       ^       1 L 

J  (ax2  +  6)2         2  a  ax-'  H 


+  6 

23.  f^     f      ^  = ^ +  J-  (•     '^■^-      ;  then  18. 

J  (rtx2  +  by      2  &(«x2  +  fe)       2  6  J  «x2  +  &  ' 

24.  C     r^\,    =ro^f-1  ;  then  7  or  8. 
J{ax^  +  b)^      L2  a  J  11^' ju  =  axni 


f       <^a:        _        J X 2  }>i,  -  3     C  dx 

J  (ax2+ft)"'     2ft(m-l)  (fflx2  4-&)"-i      2(m  -  1)6  J  (ax2  + M".-i 


25. 

2(m-l)6J  (ax2  +  6)' 
Notes,     (a)  Use  25  repeatedly  to  reach  23  and  thence  18. 
(ft)  Final  forms  in  partial  fraction  reduction  are  of  types  12,  24,  25  (by  use  of  21). 


IV,  C]  IRRATIONAL   ALGEBRAIC  37 


C.    (a)  Integrand  Irrational :  involving  /•  =  \  aj-  +  b. 

26.    I   \  F  {X,  Vajc  +  6)  dj-  \        ^ =  ( r( '"' "  ^ ,  r]^  dr, 

2r 


\  Vrtx  +  h  d.r  =  \  r—  dr  =  —  r',  r  =  y/ax  +  b. 
J  J      a  3  a 

(.>■  Vax  +  6  d.r  =  -( (?•*  -  bf^) dr  =  —  [--^'\. 
J  d^  J  d^  \_b      3  J 

f      ^'       ='Udr  =  '^r. 
-^  Vax  +  b      a  J  a 


30.     r dx ^  r^^r  ;  use  9  or  10. 

J  .■  ->/„..  A.h     J  r-  —  b 


Vox 

Note.    Vi/a-  +  //  ==  ((/.i-  +  ?<)  V(Z7+7, ;    (%/,/,»•  +  /,)3  =  (^ax  +  b)  v'<fx  +  b. 


(b)  Integrand  Irrational :  involving  V±x^  ±  a'^. 
32     r   ,— —^  =  are  sin  •'"=sin-i''  =-008-1"'  +  [coust.l. 
33.    f     ^^       =  log  (J?  +  y/jc^Ta'i)  =  sinhi  -  [  +  const.]  for  +, 

or  cosh-i  -  [+  const.]  for  — . 


_    C da;         ^  ^■^_,  /-liiLJ^U  -  cos-i  ^-^li^  r  +  const.] 

-^  \/2^  -  a;-^  \    a     J  a 

=  vers-i  X  +  const. 

.     f ^^g =  lsec~^~  =  ^  cos-i  "  =  -  icsc-i- [+  const.]. 

•^  '.-  lAri  _  /,2      a  a      a  x         a  a 


xVx^-d^      «  a      « 


3g     r^^=_V^2Z:^2.         38.    ra:Va2-x^dx=-:^(Va2-x2)8. 

37.    r     ■^(^■''       =Va:2  +  qg.  39.    fx  Vx2  +  a'^  (?x  =  j  ( Vx^  +  n^)». 

Notes,     (r/)  32  and  3.1  furnish  the  basis  for  all  which  follow. 

(b)  3G,  3T,  3»,  89  follow  from  xdx  =  </(a;»  +  const.)/2. 


38  STANDARD  INTEGRALS  [IV,  C 

40.    r     ^'dx      ^_gv^?ir^^^g'sin-ig. 


41.    f— ^^=_llogr^±^^^!A^]. 
•^  xVa^  ±  a;2         a       L  x  J 


42 


.     r ^-^  =  - -L  Va-^  -  x2.        43.   Va2_a;2 


Notes.     (,()    rV^TTZir!  ^a;  =  a^  C       '^"'        -  f     '^''^"'      ,  then  32,  40. 
(to  43)  J  J  Va^  -  ib2     J  V^J  _  iB2 

(?/)    t  dx  =  ai  I  — ,  -  i  — ^=^r ,  then  36,  41. 

('■)    r^:^«if^,7a,=  a2  (•         ^^  -  r^^.  then  42,  32. 
♦^         «■  J  a;2Va2  -  xi     J  Vai  -  x^ 

44.     r     ^'^^       =  ^  V^2^^2  ^  ^'log  (X  +  yiW±~a'). 
■^  y/x'-  ±  a"'^      2  2 


45 


J  x2  Va;-^  ±  rt"-^  «^^  Vx^  ±  a2 


Notes,     (a)   f  Va-^  ±  u^  c/a;  =  f    "^     "^     ±  «=  J"  ■ ,  then  44,  33. 

(to  46)  "■  ■^Vic2±a2  ■'v'ai2±a2 

(6)  J tto^  [— •+ogJ ^.  then  37,  35,  or  41. 


r      <?x       _       X  4g   r      (7x       _ 


±a; 


)3      a2  Vx2  ±  a2 

Notes.     Trigonometric  Substitutions.     If  the  desired  form  is  not  found  In  32-48,  try 
79.     Then  use  Nos.  55-79,  see  79.     {h)  See  also  D,  51-54,  below. 


49.   V±(ax2  +  6x  +  c)=v^^±  ^x  +  JlV^^1^^^ 


Notes.    Forms  containing  V ±(aaj2  +  te  +  c)  : 

(a)  \  F{x,  V±(a!B2  +  &a!  +  c)  dee,  —  use  49,  then  put  m  =  a;  +  ^-/'^  a  [see  49(6)];  then 
use  32-18. 

(6)  Remember  v/±(aa-2+ te  +  c)'  =  ±(rta-2  +  hx  +  c)  v/±(rta-2  +  fta- +  c). 

Vi  (t/a-s  +  bx  +  c)  =±(aa2  +  to  +  c)-=-v'±(«a'»  +  6a;  +  c). 
(c)  Simplify  all  radicals  first. 


IV,  E]  REDUCTION   FORMULAS  39 

ax  +  b 


50    ^/«£±J?  _  ax  +  ft v'(a.c  +  b){cx  +  d) 

^<^x  +  d     V(ax-hb)(cx4-d)  ex  +  d 


Notes.    (</)  Integrals  containing  V^^a-  +  l>)/(ar  +  d) :  use  50,  tlien  49,  then  32-48. 
(b)  Substitution  of  «  =\/ {aoe  +  b)/{cx  +  d)  is  successful  ^vithout  50. 

D.    Integrals  of  Binomial  Differentials  —  Reduction  Formulas. 

Symbols :  «  =  ax"  +  6  ;  a,  ft,  p,  m,  n,  any  numbers  for  which  no  de* 
nominator  in  the  formula  vanishes. 


52 


.    (x"\ax''  +  b)Pdx^ ? [x'^+^up  +  7ipb  (x"*uP-^dxV 

J  in  +  np  +  1  J 

\x^(ax^  +  b)P(lx 

=  — — ! [-  x'"+hip+^  +  (m  +  n  +  np+l)  (x"'uP+^  dx]. 

bn{p  +  1)  J 

53.    (x^'^iax^^  +b)Pdx 

= — -[.v^+ij/p+i  —  a(m  +  n  +  np  +  1)  (x^^^uPdxl. 

(?n-|-l)ft  J 

64.    (xr*^{ax'^  +  h)Pdx 

=  — ^- ^[.r'"-"+i?t/'+i  -(7)1  -  71  +  l)ft  rj7»»-"MPrfx]. 

a{rn  +  np  +  1)  J 

Notes,    (a)  These  reduction  formulas  useful  when  p,  m,  or  n  are  ft-actional; 
hence  applications  to  Irrational  Integrands. 

(fi)  Repeated  application  may  reduce  to  one  of  32-48. 

(c)  Do  not  apply  if  j?,  m,  ?),  are  all  integral,  unless  ?!  ^2  and  p  large.     Note  11,  15,  17-25w 

En.    Integrand  Transcendental :    Trigonometric  Fiinctlons. 

55.  I  sinu^rfa?  =— cos  J7. 

56.  Tsin'^  xdx=—  ^  cos  x  sin  x  -f-  ^  x  =  —  J  sin  2  x  +  \x. 
Note.  J"  sin*  kx  dx,  —  set  kx  =  u,  and  use  56.    Likewise  in  55-78. 

57.  rsin»xdx=-?inri£^2^  +  !Lziirsin»-2xdx. 
J  n  n     J 

Note.     If  n  is  odd,  put  sin' a;  =  1  —  cos' a;  and  use  62. 


'xdx. 


40  STANDARD  INTEGRALS  [IV,  E 

58.  (cosxdx  =  8mx. 

59.  i  cos-xdx  =  ^sinxcosx  +  ^  a;  =  ^  sin2  a;  +  i*. 

60.  (cOSnxdX  =  ""^"-^  ^  '^'"  ^  +  'J—^    (cOSn-^xdX. 

J  n  n     J 

Note.     If  n  is  odd,  put  cos^a;  =  1  -  sin^sc  and  use  63. 

61.  j  sinxcosxfZx  =  —  ^cos2x  =  ^sin2x[+  const.]. 

62.  f  sin  x  cos^xdx  =  -  ^"^""^^  ^ ,  n  9^  -  1. 
•/  n  +  1 

63.  rsin'>xcosxdx  =  ?i5!^    „^_  1. 

J  71+1 

64.  fsiu"  X  COS"  xdx^^'""^'^-""^"-^  ^  +  ^^1^1-1  fsinnxcos-^^dx 

— sin«— 'xcos™+ix  ,   w— 1  T  . 
m  +  n  m+  nJ 

Note.     If  n  is  an  odd  Integer,  set  sin'  a-  =  1  -  cos'  x  and  use  62.     If  ;»  is  odd,  use  ( 

65.  fsin  (mx)  cos  (nx)  dx  =  -  ^""  ^^^^  +  "^  '"J  -  ^""^^"^  "  ^^^^^ , 
J  2(m+?0  2(m-?i) 

66.  fsin  Onx)  sin  (n,r)  dx  =  ^^^I0ri-n)x]  _  sin[(»i  +  >t)x] 
»^  2(m  -  ?i)  2(ra  +  71) 

67.  rcos(77ix)cos(7U-)(Zx^^^"'^("^-  "^■'"-1  +"'"l^("^  +  ")-'^J,  m=^±n. 

•'  2(»?l  —  ?l)  2(771  +  71) 

68.  j  tan  xdx  =  —  log  cos  x.  69.    ftan"-  x  dx  =  tan  x  —  x. 

70.  ftan^x^x  =  ^^""~^^-  _  C tsin''~2 ^.4^.,^ 
J  71  -  1         J 

71.  j  ctii  X f?x  =  log  sin  x.  72.     rctn2  ,/•  dr  =  —  ctn  x  —  x. 

73.  (ctW'xdx  =-^^""~'^-  rctn"-2.rd>:. 

74.  Jsec  X  dx  =  log  tan  ( f  +  7  j  =  log  (s^c  x  +  tan  x)  [  4-  const.  ], 


m  ^  ±  n. 


IV,  E]  TRIGONOMETRIC  4] 

75.  (  CSC  X  dx  =  log  tan  ?  =  —  log  (esc  x  +  ctn  x)  [  +  const.  ]. 

76.  fsec-  xfix  =  tan  x.  77.     (cac'^xdx  =  —  ctn  z. 

TseC"  X  CSC"  a;  dr  =  \ ~ (See  also  04.) 

J  J  sin"  X  COS""  X 

m-l      J 


78. 


?ft  —  1 

1 


sec'"""'^  csc"~'  .r 


sec'"--xcsc"xdx 


■  .sec"-i  X  csc"-ix  +  "'  +  "  -  ^  fs 
n  —  1      J 


'  X  csc"-2  X  dx. 


n-  1 

Notes,     (a)  In  64  and  TS  and  many  others,  m  and  7^  may  have  negative  values. 
(6)  To  reduce  J[sin"ir/cos»" a-]  t/«  take  7n  negative  in  64. 
(c)  To  reduce  J"(cos»"a'/siu"a']rfa;  take  n  negative  in  64. 

79.    SubatitutioBB  : 


1    u  = 

dti 

sinx 

C08X 

tanx 

X 

dx 

<„ 

sincT 

cos  a;  cfa; 

u 

v/1  -  «J 

« 

sin-»!t 

(/!/ 

V.  -  »J 

Vi  -«t 

cos  a- 

—  sin  X  dx 

Vl  -  h2 

M 

Vl   -  «2 

cos-i u 

rf« 

Vi  _  u^ 

(3) 

tan  X 

ffctxdx 

« 

1 

« 

tan-'  « 

du 
!  +  «» 

Vl  +  «2 

Vi  +  «2 

W 

sec  3!  tan  x  dx 

V>/'-l 

1 

Vh2  _  1 

see-'  K 

du 

«V«i-  I 

!  '"" ;; 

l.e.^-dx 

2u 

1    +  «2 

1  -  »/J 

1  +  «J 

,-«a 

1  +  «2 

Replace  ctn  x,  sec  x,  esc  x  by  1/tan  x,  1/cos  x,  1/sin  x,  respectively 
Notes,    (a)  J  /"(sin  x)  coaxdx,  —  use  79,  (1). 

(b)  /  F(cos  35)  sin  x dx,  —  use  "9,  (2). 

(c)  J'/f(tana-)sec»a!da-,  — u8e-9,  (3). 

(d)  Inspection  of  this  table  shows  de»irahle  ttubsiUutionii  from  trigonoiiietrio  to 
algebraic,  and  conversely.    Thus,  if  only  tan  z,  sin'z,  cos' a;  appear,  use  79,  (8). 


42  STANDARD  INTEGRALS  [IV,  E 

80.    r ^ =         ^         sin-i^  +  ^"'"^. 

^  a  +  6  sin  X      Va'^  —  6^  a  +  h&mx 


1         i^gL=j4!^,^i±^tan(2Z21,i{«2<52. 


Vft^  _  a2        5  +  V62  -  a'-^  +  a  tan  (a;/2) 

81.    r ^ = 2 ^^^_,  r  J^36  ^^^  q ,  «2  >  62 . 

Ja  +  bcosx      V(j2  _  52  L^a  +  &        2  J 


1        ipg  V^  +  g  +  V^  -  g  tan  (x/2)  ^  ^2  ^^3 


V62  _  a^         V6  +  a  —  -v/fc  —  a  tan  (x/2) 

.    f ^ =         1         logtan^±^.  «  =  sin-i        ^        ■ 

J  a  sin  a;  +  6  cos  X      Va^  +  h'^  2  Va"^  +  6^ 


•>  a  +  b  &mx  ^  a  +  h  s\ax     b  ^  b        •'  u+Z»  sin  as 

then  use  82  a,  80. 

(c)    Many  others  similar  to  (a)  and  (6);  e.y.  J[sin«/(a  +  &  cosa;)]  dte,  —  use  V9,  (2). 

id)  f   ,   „    ^"',, ;-  and  like  forms,  —  use  79,  (3)  ;  see  79,  note  d. 

^   ^  J  o2  sin2  X  +  h^  cos2  x 

(«)  As  last  resort,  use  79,  (5),  for  any  rational  trigonometric  integral. 

Eft.  Integrand  Transcendental :  Trigonometric-Algebraic. 

83.  \  x™  sin  xdx=—  x"'  cos  x  +  to  j  x™-'^  cos  x  dx. 

84.  \  x™  cos  X  dx  =  x™  sin  x  —  to  i  x"'-i  sin  x  dx. 

Notes,    (a)  /  aj  sin  tc  c?a)  =  —  «  cos  a?  +  J  cos  a;  dx,  —  use  58. 
(6)  /«"'  sin  vidx,  — repeat  83  to  reach  58. 

(c)  J  (Any  Polynomial)  sin  »  rfa;,  —  split  up  and  use  88. 

(d)  For  cos  SB  repeat  («),  {h),  (c). 

35     rsinxdx^       -sinx       ^  _J_  fcosx^^^  ^  _^  ^^ 

J        X™  (to  —  1 )  X™-!        TO  —  1  J  X™-! 

86.    r52?^^  = ^os^ L_  r!i5^dx,  TO^l. 

J      x"  (m  —  1)  x™-!      TO  —  1  J  x™-! 


IV,  E]  TRANSCENDENTAL  43 

87.  ^•^rfx=j'[l-^  +  ^-...]d.c;seeII,E,13,p.^. 

88.  f^2if,,.,^rrl_^  +  £!_...l,;^.seeII,E,14,p.5. 
J     X  J  Lx      2 !      4 !  J 

Note.    Other  tiigouoinetric-algebraic  combinations,  use  5 ;  or  "9  followed  by  89-94. 

Ec.    Integrand  Transcendental :    Inverse  Trigonometric. 

89.  I  siii-i  xdx  —  x  siu-i  x  +  Vl  —  x^.     [From  5.] 

90.  I  cos-i  xdx  =  x  cos-i  x  —  Vl  —  x^. 

91.  I  tan-i  xdx  =  x  tan-^  .r  —  |  log  (1  +  x^). 

92.  r.,"sin-ixdx  =  5*^^«"^li-^---L^  fl^-:!!^^,  then53or54,32,36. 

93.  fx"  cos-i  X  dx  =  ^""^^  '^"^"'  ^  +  -—  r-'"'^'  ^^^  ,  then  53  or  54,  32,  3d. 
J  «  +  1  M  +  1  J  Vl-x'^ 

94.  r.-"tan-ixdx^y'^'^^""'^---i-  f?!^lii?^    then  19  (c). 

J  K  +  1  ?J  +  1  J    1+  X2  '  ^    ^ 

Notes,    (a)  Replace  ctn-'a;  by  -  -  tan-' j- ;  or  by  tan-i  (l/.c)  and  substitute  \/x  =  u. 

{b)  Replace  sec~>a;  by  cos"' (I/a"),  csc"'j  by  sin-i(l/^)  and  substitute  \/x  =  u. 
(c)  J(.^ny  Polynomial)  sin-^xdx,  split  up  and  use  92.    (Similarly  for  cos-'sr,  etc.) 
('')  J/W  sin~»«rfa;,  -  use  (.5)  with  u  =  sin-»a5.    (Similarly  for  cos"iiB  and  tan-»(w  ) 
(#)  Other  Inverse  Trigonometric  Integrands,  use  79  or  5. 

Erf.     Integrand  Transcendental :    Exponential  and  Logarithmic 

95.  Ca'dx  =  -*^^  =  ~^-  logiQ  e  =  ,   "—  0.4343. 
J  logea     logio«  logio« 

96.  \e''ilx  =  e^. 

Notes,  (a)  J e^^dx  =  <*'-=-*.  (i)  Notice  a*  =  eClog*")*  =  e*',  A  =  log.a. 

97.  rx"e*^df  =  -x^e^  -  "  (x^-^e^Ulx. 

Notes,     (a)  J  Lr«*^</a-  =  ire'^/yl-  -  el^/iK  (b)  Ja-«e**(7a',  —  repeat  97  to  reach  97  Cn> 

(c)  /  (Any  Polynomial)  e^'dx,  split  up  and  use  97. 


44  STANDARD  INTEGRALS  [IV,  E 

98     C^  dx= —- \ ^  T-l^  dx  (repeat  to  reach  99). 

J  a;"*  (m  —  1)  x"*"!      m  —  1  J  a:"*"^ 

99.  C— (Zx  =  (  [I  +  1  +  -^  +  —  +  --Idu,  ti  =  kx  ;  see  TaftZes,  V,  H 
J   X  J   Lu  2 !      ;J !  J 

,  /^/^     r  -^   •  7         fc,  A  sin  nx  —  n  cos  «x 

100.  \  e*^  sin  nx  dx  =  e*'' — — — . 

J  K^  +  rfi 

,M     C  1,^  7         kx  ^  COS mx  +  m  sin  mx 

101.  I  e''^  cos  m.r  (Zx  =  e** r-— ! ; . 

102.  I  log  X  dx  =  X  log  X  —  X. 

103.  i:(\ogxY'^  =  ^^^^&^y^,n^-l. 
J  X  n  +  1 

104.  C^^  =  r^^' ,  tt  =  log  X  ;  see  99  and  TaUes,  V,  H. 
J  log  X      J     u 

105.  f  .X"  log  X  dx  =  x"+i  rl^g^ ^-1  . 

106.  fe**  log  X  fZx  =  -  e*»  log  x  -  J  f  —  cZx,  see  99. 
J  k  k  J  X 

F.   Some  Important  Definite  Integrals. 

107.  i     -^^  =  — - —  ,  if  »W  >  1  (otherwise  non-existent). 
J\  xm     m—l 


108.  r      ^at        ^  _JL . 

Jo  a2  +  fc2a;2      2  a6 

109.  I    icwe-*  rta?  =  r  (rt  +  1)  =  n !  if  n  is  integral.   See  V,  F,  p.  54. 


(a)  In  general,  r  (w  +  ] )  =  m  •  T  (n)-  as  for  »i  !,  if  n  > 0. 

(?.)  r  (2)  =  r  (1)  =  1,  r(iA')  =  Vn.    r  («  + 1)  =  n  («). 


110.  rxni-.r)"dx  =  I^(»^  +  ^>^(»  +  ^>. 
>  r  (m  +  n  +  2) 

111.  i    sinnoc  ■  smnixdx  =  \    cosna'cos'inxdx-—0,  \{  tn^^n,   ^ 

Jo  Jo  IT-, 

if  m  and  ?i  are  integral. 

112.  I    sin2  nx  dx  =  \     cos'-  nx  dx  =  7r/2  ;  n  integral,  see  .5G,  59. 
Jo  Jo 


IV,  G]  DEFINITE  INTEGRALS  45 

113.    f"e-*^dr  =  l/*-  114.     P[(sinnj)/j]dx  =  7r/2. 

115.  pe-*'  sin  nx  dx  =  n/^k'^  +  n-),  if  k  >  0. 

116.  P e-*'  cos  mx  dx  =  A•/(^-  +  m-) ,  if  A-  >  0. 

117.  Ce-^^x"  dx  =  ^^  ^^^  "^  ^^  =  ^^  ,  if  n  is  integral.     See  109. 

118.  Pf^-*'^^^.r  =  V^/(2^•)• 
119.    J^  e-^^^=  cos  mx  dx  =  -^-r ,  if  ^•  >  0. 

120.    r-ii?^-=  P— ^^-  =  — •      121.     r(loga-)«d:<;  =  (-l)»n! 
Jo  e*r4.e-Ai     Jo  cosli  ix     2  k  Jo 

122.  p^'log  sin  a:  dx  =  p  'log  cos  x  dx  =  -  |  log  2. 

123.  r'"''sin-^"+i xdx  =  P''"cos2»+' xdx  =  „   !'t'^"'^",    («>  positive 
Jo  Jo^  3.5.7...(2n  +  l)^    ^^^^^^^  ^ 

124.  ("'"'■sin^" xdx  =  (""'"cos^" xdx  =     ' '  '   /"/"""  "^^  ^  ("'  positive 
^'  ^^  2.4.b...2«        2       jj^^^g^^^ 

G.   Approximation  Formulas. 

125.  Cf(x)dx=f{c)(b-a),     a<c<b.    [Law  of  the  Mean.] 

126.  Cfix)dx  =  /(^^  +  /(^)  (6-0).       [Trapezoid  Rule  — precise 

for  a  straight  line.] 

127.  Cf(x:)dx.  [Extended  Trapezoid  Iinle.'\ 

=  [f{a)/2  +  f(a  +  Ax)+f(a  +  2Ax)+  ■■■  +/[a  +  (n-l)Ax]+/(6)/2]Ax. 

128.  ["/(x) dx  =  /(a)-f4/[(.  +  ^)/2]  +  /(fe)  ^^  _  ^^^ 

[Prismoid  Rule  ;  or  second  Simpson- Laf/range?ipprox\ma,t\on  ;  precise 
if  /(x)   is  any  quadratic  or  cubic  ;  see  §  71,  p.  12(5.] 

129.  Cfix)dx  =  ^  [/(a)  +  4/(rt  +  Ax)  +  2/(«  +  2  Ax) 

+  4/(a  +  3  Ax)  4-  2/(a  +  4  Ax)  +  ••.  4-/(6)]. 
[Simpson's  Rule  ;  or  extended  prismoid  rule.     See  §  125,  p.  240.] 


46  STANDARD  INTEGRALS  [IV,  G 


|V(x)( 


130.    \   f(x)dx 


=  /(«)  +  3/ra  +  Ax]+3/[a  +  2Ax]  +  /(ft)  ^j,  _  ^y,  Ax  =  (b  -  a)/3. 
8 
[A  third  Simpson-Lagrange  Approximation.     Extend  as  in  129.] 

131.  r  /(a-)  dx 

-!/(„)  + 82/la  +  Ax]+ 12/[«  +  2  Aa;1+  32/[ffl  +  3  Ax\+  7.f(b)  (^      ^^.  Aa,  =  (,,_rt)/4, 

lA  fourth  Simpson-Lagrange  Approximation;  see  Lagrange  interpola- 
tion formula,  II,  I,  17,  p.  15.'] 

H.  Standard  Applications  of  Integration. 

132.  Areas  of  Plane  Figures :    \  dA. 

(a)  Strips  AA  parallel  to  y-axls  :  dA  =  y  (bst: 

(6)  Strips  A  A  parallel  to  aj-axis :  dA  =  x  dy. 

(fi)  Rectangles  A  A  —AxAy:  dA  =  dx  dy,  A  =^j  dx  dy. 

{d)  Parameter  form  of  equation:  A  =(l/'2) /(a-rfy  -  ydx). 

(e)  Polar  sectors  bounded  by  radii  :  dA  =  {p^/2)  dd. 

(/')  Polar  rectangles  A,4  =  p  Ap  AS :  dA=pdpde;  A  =^  ^pdp d0. 

133.  Lengths  of  Plane  Curves  :    \  ds. 

(a)  Equation  in  form  y  =/(aj)  :  ds  =  Vl  +[f(x)fdx. 
(6)  Equation  in  form  x  =  ^{y):   d.<<  =  \^1 +[<}>' (y)\'dy. 

(f)  Parameter  equations  :  ds  =\/ dx^  +  dy^. 
(d)  Polar  equation  :  ds  =\^J^~+~^d9^. 

134.  Volumes  of  Solids :    (dF. 

(a)  Frustum  (area  of  cross  section  A):  dV=  Adh\   V  =  iAdh  where  h  is  the  variable 
height  perpendicular  to  the  cross  section  A . 

(6)  Solid  of  revolution  about  se-axis  :  dV  =  ■ny''-  dx. 

(c)  Solid  of  revolution  about  y-axis  :  d  F  =  -n^dy. 

(d)  Rectangular  coordinate  divisions  :  dV=dxdydz\ 

(e)  Polar  coordinate  divisions  .  dV=-  p^  sin  9  dp  d<^  dO. 


IV,  H]  APPLICATIONS  47 


135.   Area  of  a  Surface :    J  jsec  «|»  dx  dy, 


where  \{/  is  the  angle  between  the  element  ds  of  the  surface  and  its  pro- 
jection dxdy. 

((/)  Surface  of  Kevolution  about  te-axis  :  .4  =  J  2  iry  ds. 

{b)  Surface  of  Revolution  about  y-axis:  A=l'2irxds. 

136.  Length  of  tioisted  arcs :      \  ds, 

(n)  Kectangular  Coiirdiuatcs  :  d-t  =  N/rfa-J  +  dy^  +  dzi. 

(6)  Explicit  Equations  i/  =/(x),  s  =  <f,  (,r)  :    ds  =  Vl  +  [/'  W)?  +  [*'  (a-)]'. 

(c)  »  =/(0,  y  =  *(0,  2  =  «A(0  :  <^«  =  '^U'W^+WiW+WW'- 

(d)  Polar  Coordinates  :  ds  =  VrfpS  +  ptd<f)^  +  (fi  cos>  0  dO». 

137.  itfass  0/ a  body :     3I=(dM=(pdV, 

where  p  is  the  density  (mass  per  unit  volume). 

(a)  Ifp  is  constant:    Jr=pidr;  see  13-1. 

(b)  On  any  curve  :     dV=ds,ifp  =  mass  per  unit  length. 

(c)  On  any  surface  (or  plane) :    dV=  dA,  if  p  =  mass  per  unit  area. 

138.  Average  value  of  a  variable  quantity  q  :     A.  V.  of  q.  : 

(a)  throughout  a  solid  :     q  =/(.x,  y,  z)  \  A.  V.  of  q.  =  p2  dV-r-  J  rf  T. 
(6)  on  an  area  A:    A.  V.  of  q.  =  ^qdA  ^ ^dA. 
(c)  on  an  arc  « :     A.V.  of  q.  =  jqds  -i-  Jds. 

139.  Center  of  Mass,    (x,y,~z):    x=\xdM-^(dM, 
with  similar  formulas  for  y  and  z.    See  dM.,  137. 

(a)  for  a  volume  :    dJf=  pd  F. 

(b)  for  an  area :        rfJ/"=  p  dA. 

(c)  for  an  arc  :  dM  =  pds. 

139.*    Theorems  of  Pappus  or  Guldin  : 

(a)  Surface  generated  by  an  arc  of  a  plane  curve  revolved  about  an 
axis  in  its  plane  =  length  of  arc  x  length  of  path  of  center  of  mass  of  arc, 

(h)  Volume  generated  by  revolving  a  closed  plane  contour  about  an 
axis  in  its  plane  =  area  of  contour  x  length  of  path  of  its  center  of  mass. 

1^0.    Moment  of  Inertia:     I=(r'idM.     (See  137,  139.) 

(a)  For  plane  figures.  Tz+ Ty=  T„,  where  Tx,  I\i.  h  "re  t.iken  about  the  tr-axis,  the 
j/-axis,  the  origin,  respectively. 

(ft)  For  space  figures,  /i  +  Iy+  Im=  /o- 

(o)  /,  =  /-  +  («-  S)'Jf,  where  1^  is  taken  about  a  lino  ||  to  the  xaxis. 


48  STANDARD  INTEGRALS  [IV,  H 

141 .  Badius  of  Gyration :    k'i  =  I -i-  M  =  TrS  dM  -4-  (dM. 

[In  140  and  141,  r  may  be  the  distance  from  some  fixed  point,  or  line,  or  plane.] 

142.  Liquid  pressure  :     ^^  =  i  p^  dA^ 

where  p  is  the  total  pressure,  dA  is  the  elementary  strip  parallel  to  the 
surface  ;  h  is  the  depth  below  the  surface  ;  and  p  is  the  weight  per  unit 
volume  of  the  liquid. 

143.  Center  of  liquid  pressure  :    h=  \  Ji^dA  -^  i  hdA, 

•     144.    Work  of  a  variable  force  :     W  —  \f  cos  +  ds, 

where /is  the  numerical  magnitude  of  the  force,  ds  is  the  element  of  the 
arc  of  the  path,  and  \p  is  the  angle  between  /  and  ds. 

145.    Attraction  exerted  by  a  solid:    F=  fef^^^^^, 

where  k  is  the  attraction  between  two  unit  masses  at  unit  distance,  m  is 
the  attracted  particle,  dM  is  an  element  of  the  attracting  body  ;  r  is  the 
distance  from  m  to  dM. 

Components  F^,  Fy.  F^  of  F  along  0«,  Oy,  Oz  are  : 


Fx  =  kmj — ,    Fy  =  kmj ■,    Fz  =  k>n^ -^ ,    . 

■where  a,  3,  y  are  the  direction  angles  of  a  line  joining  )n  to  dJ/. 

146.  Work  in  an  expanding  gas :     W  =  \p dv. 

147.  Distance  s,  speed  v,  tangential  acceleration  jr: 

JT=  ijvdt^  i  I  CsdtXdt. 
[Similar  forms  for  angular  speed  and  acceleration.] 

148.  Errors  of  observation : 

y  d.r,  where  y  is  the 
J, „ ^. „  „.  „-j,"" — -  -•  ~" 

(b)  The  usual  formula  y  =  (/t/Vi^)  e"*'^"  gives:    P  =  (/i/Vn)  j  e '''-''■  dx,  where  h 
is  the  so-called  measure  of  precision. 

(c)  Probability  of  an  error  between  x  =  —  a  and  x  =+  a:    P  (a)  =1  y  dx. 

(d)  Probable  error  =  (0.411) /h  =  value  of  n  for  which  P(.a)  =  1/2. 

(e)  Jfean  error  =  C    xydx-r-  C    ydx  =  \/(liV7) 


V.   NUMERICAL   TABLES 


A.  TRIGONOMETRIC   FUNCTIONS 

[CharacU'ristics  of  Logaiithuis  omitted  — deteriuine  by  the  usual  rule  from  the  vahu' 


i:a..ia,.s 

De 

Sine 

Tangent 

I'OTANGEXT 

NK     1 

-rees 

Value  loBxo 

\alue  lo^'io 

Value 

log.o 

Value 

'uV'io ! 

0000 

0° 

.0000  -X 

.0000  -00 

00 

00 

1.0000 

0000  !X)° 

1.5708 

.0175 

1° 

.0175  2419 

.0175  2419 

57.2SX» 

7581 

.99<t8 

9999  i  89° 

1.5533 

.0.U9 

2° 

.0349  5428 

.0349  54.31 

28.()36 

4569 

.9994 

9997  88° 

1.5.359 

.0024 

.r 

.0523  7188 

.0.-.24  71'.>4  19.081 

2806 

.998(5 

9994 

87° 

1.5184 

.0{;98 

4"^ 

.0(j98  8436 

.(Xi'.H)  8446 

14.301 

1554 

.9976 

998!' 

86° 

1.5010 

.0873 

5° 

.0872  9403 

.0875  9420 

11.4:?0 

0580 

.i>9()2 

9983 

85° 

I.IS.T. 

.1047 

li" 

.1045  0192 

.1051  0216 

9.5144 

9784 

.9i>45 

imc, 

84° 

i.4(i(;i 

.1222 

7° 

.1219  0859 

.1228  0891 

8.1443 

9109 

.9925 

9i)(58 

83° 

1.44.S(i 

.1396 

8° 

.1392  1436 

.1405  1478 

7.1154 

8522 

.9903 

{)958 

82° 

1.4312 

.1571 

9^ 

.15(>4  1943 

.1584  1997 

6.3138 

8003 

.9877 

9946 

81° 

1.4137 

.1745 

10° 

.1736  2397 

.1763  24<)3 

5.(5713 

7537 

.9848 

9934 

80° 

1.3963 

.1920 

11° 

.1908  2806 

.1944  2887 

5.1446 

7113 

.9816 

9919 

79° 

1.378S 

.2094 

12° 

.2079  3179 

.2126  3275 

4.7046 

6725 

.9781 

95t04 

78° 

1.3614 

.2269 

13° 

.2250  3521 

.2309  36;U 

4.3315 

6;«56 

.9744 

9887 

77° 

1..34:'.!i 

.2443 

14° 

.2419  3837 

2493  3968  4.0108 

6032 

.9703 

9869 

76° 

1.. 32(55 

.2618 

15° 

.2588  4130 

.2679  4281 

3.7.321 

5719 

.9659 

9849 

75° 

1..3090 

.2793 

16° 

.2756  4403 

.28(57  4575 

3.4874 

5425 

.9(513 

9828 

74° 

1.2915 

.2t)67 

17° 

.2924  4659 

.3057  4853 

3.2709 

5147 

.9563 

9806 

73° 

1.2741 

.3142 

18° 

.3090  4900 

.3249  5118 

3.0777 

4882 

.9511 

9782 

72° 

1.2566 

.3316 

19° 

.3256  5126 

.3443  5370 

2.9042 

4630 

.9455 

9757 

71° 

1.2392 

.3491 

20° 

.3420  5311 

.3640  5611 

2.7475 

4389 

.9397 

9730 

70= 

1.2217 

M65 

21° 

.3584  5543 

.3839  5842 

2.6051 

4158 

.933<5 

9702 

69° 

1.2043 

.3840 

22° 

.3746  5736 

.4040  60()4 

2.4751 

3936 

.^)272 

9672 

68° 

1.1868 

.4014 

23° 

.3907  5919 

.4245  6279 

2..35.59 

3721 

.9205 

9(540 

(57° 

1.1(594 

.4189 

24° 

.4067  6093 

.4452  6486 

2.2460 

3514 

.9135 

9607 

6<)° 

1.1519 

.4363 

25° 

.4226  6259 

.46(i3  6687 

2.1445 

3313 

.9063 

9573 

(55° 

1 .1.345 

.45:38 

26° 

.4384  6418 

.4877  6882 

2.0503 

3118 

.8988 

9537 

64° 

1.1170 

.4712 

27° 

.4540  6570 

.5095  7072 

1.9626 

2928 

.8910 

9499 

63° 

1.099(5 

.4887 

28° 

.4695  6716 

..5317  7257 

1.8807 

2743 

.8829 

9459 

62° 

1.0821 

.5061 

29° 

.4848  685(i 

.5543  74;?8 

1.8040 

25(52 

.874(5 

9418 

61° 

1.0(547 

.5236 

30° 

..5000  6«)0 

.5774  7614 

1.7321 

2.386 

.8660 

9.375 

(50° 

1 .0472 

..5411 

31° 

.5150  7118 

.(5009  7788 

1.6643 

2212 

.8572 

9331 

59° 

1.0297 

.5585 

32° 

.5299  7242 

.()249  7958 

1.(5003 

2042 

.8480 

9284 

58° 

1.0123 

.5760 

33° 

.5446  7361 

Ami   8125 

1.5399 

1875 

.8387 

92m 

57° 

.{•948 

.5934 

34° 

.5592  7476 

.6745  8290 

1.4826 

1710 

.8290 

9186 

5(5° 

.9774 

.(;109 

35° 

.5736  7586 

.7002  8452 

1.4281 

1548 

.8192 

9134 

55° 

.9.-99 

.()283 

3<)° 

.5878  7692 

.7265  8613 

1.37(54 

1387 

.80iX) 

9080 

54° 

.9425 

.6458 

37° 

.6018  7795 

.7536  8771 

1.3270 

1229 

.7986 

9023 

53° 

.9250 

.6632 

38° 

.6157  7893 

.7813  8928 

1.2799 

1072 

.7880 

8(H5.5 

52° 

.907(5 

.6807 

39° 

.6293  7989 

.8098  9084 

1.2319 

0916 

.7771 

8i)05 

51° 

.8901 

.r,981 

40° 

.6428  8081 

.8391  92.38 

1.1918 

0762 

.7660 

8843 

50° 

.8727 

.7156 

41° 

.6561  8169 

.8693  9392 

1.1501 

0608 

.7547 

8778 

49° 

.8552 

.73:«) 

42^ 

.(i691  8255 

.9004  9544 

1.1106 

04.5(5 

.7431 

8711 

48° 

.8378 

.7505 

43° 

.6820  8338 

.9325  96tt7 

1.0724 

0303 

.7314 

8641 

47° 

.8203 

.7679 

44° 

.6917  &418 

.9657  9848 

1.0355 

0152 

.7193 

8.569 

4(5° 

.8029 

.7854 

45° 

.7071  8495 

1.0000  0000 

1.0000 

0000 

.7071 

8495 

45° 

.7854 

Value  lo<r,„ 

Value  lo-,„ 

Value 

l-Pio 

Value 

lopio 

De- 

Radians 

CoSIKK 

("OTAMiENT     TaNi; 

ENT 

Sine 

srrees 

49 


50 


NUMERICAL   TABLES 


[V,  B 


B.     COMMON  LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

42 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

38 

12 

0792 

0828 

08(J4 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

35 

13 

1139 

1173 

120ti 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

32 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

30 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

28 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

26 

17 

2304 

2330 

2355 

2380 

2405 

24:30 

2455 

2480 

2504 

2529 

25 

18 

2553 

2577 

2tJ01 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

24 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

22 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

20 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

19 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

18 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

18 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

17 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

16 

27 

4314 

4330 

4346 

4362 

i378 

4:^93 

4409 

4425 

4440 

4456 

16 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

15 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

15 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

14 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

14 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

13 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

13 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

13 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

12 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

12 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

12 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

11 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

11 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

11 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

10 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

10 

43 

6335 

6345 

6355 

6:365 

6375 

6385 

6395 

6405 

6415 

6425 

10 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

10 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

10 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6(393 

6702 

6712 

9 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

9 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

9 

49 

6902 

6911 

6920 

6928 

693T 

6946 

6955 

6964 

6972 

6981 

9 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

9 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

8 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

8 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

8 

54 

7324 

7332 

7;?40 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

8 

V,B] 


COMMON  LOGARITHMS 


51 


N 

0 

1 

2 

3 

4 

3 

6 

7 

8 

9 

D 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

8 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

8 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

8 

58 

urn 

7642 

7649 

7657 

7(J(i4 

7672 

7679 

7()86 

76S>4 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

77<H} 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7915 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7iW3 

8000 

8007 

8014 

8021 

8028 

80:« 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

80<)6 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

815() 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

82;i5 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8295) 

8306 

8312 

8319 

6 

68 

8325 

8331 

83;« 

8344 

8351 

8357 

8:3<J3 

8370 

8376 

8:iH2 

6 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

6 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

6 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

6 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8(>21 

8(527 

6 

73 

8<)33 

8639 

8645 

8(>51 

8657 

8663 

8669 

8675 

8681 

8686 

6 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

6 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

6 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

6 

77 

88(>5 

8871 

8876 

8882 

8887 

8893 

88<K) 

8iX>4 

8910 

8915 

6 

78 

8921 

8927 

8932 

8938 

8943 

8»49 

8954 

8i>60 

89(>5 

8971 

6 

79 

8976 

8982 

8987 

8993 

8'.)98 

9004 

9009 

9015 

9020 

iX)25 

5 

80 

W31 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

5 

81 

;X)85 

9090 

9096 

9101 

9io<; 

9112 

9117 

9122 

9128 

9133 

5 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

5 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

i»238 

5 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

5 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

5 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

93i)0 

5 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

5 

88 

9445 

9450 

9455 

9460 

9465 

94<i9 

9474 

9479 

9484 

<>489 

5 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

5 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

958(5 

5 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

J1633 

5 

92 

9638 

9643 

9<)47 

9(>52 

9657 

9661 

9666 

9671 

9(575 

9680 

5 

93 

9685 

9689 

9(594 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

5 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

5 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

5 

96 

9823 

9827 

9832 

98:« 

9841 

9845 

9850 

9854 

9a59 

9863 

5 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

<)908 

4 

98 

9912 

9917 

9921 

9926 

t)9:«) 

i)934 

9939 

9943 

iW8 

9<)52 

4 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

•)987 

9991 

it<)96 

4 

52 


NUMERICAL  TABLES 


[V,C 


C.     EXPONENTIAL 

AND 

HYPERBOLIC 

FUNCTIONS 

e^ 

e 

-x 

siiih  * 

cosl 

X 

a, 

logea- 

Value 

logio 

Value 

logio 

Value 

logio 

Value 

logio 

0.0 

—   00 

1.000 

0.000 

1.000 

0.000 

0.000 

—  00 

1.000 

0 

0.1 

-2.303 

1.105 

0.043 

0.W5 

9.957 

0.100 

9.001 

1.005 

0.002 

0.2 

—1.610 

1.221 

0.087 

0.819 

9.913 

0.201 

9.304 

1.020 

0.009 

0.3 

-1.204 

1.350 

0.130 

0.741 

9.870 

0.305 

9.484 

1.045 

0.019 

0.4 

-0.916 

1.492 

0.174 

0.670 

9.826 

0.411 

9.614 

1.081 

0.034 

0.5 

-0.693 

1.649 

0.217 

0.607 

9.783 

0.521 

9.717 

1.128 

0.052 

0.6 

-0.511 

1.822 

0.261 

0.549 

9.739 

0.637 

9.804 

1.185 

0.074 

0.7 

-0.357 

2.014 

0  304 

0.497 

9.696 

0.759 

9.880 

1.255 

0.099 

0.8 

-0.223 

2.226 

0.;547 

0.449 

9.653 

0.888 

9.948 

1.337 

0.126 

0.9 

-0.105 

2.460 

0.391 

0.407 

9.609 

1.027 

0.011 

1.433 

0.156 

1.0 

0.000 

2.718 

0.434 

0.368 

9.566 

1.175 

0.070 

1.543 

0.188 

1.1 

0.095 

3.004 

0.478 

0.333 

9.522 

1.336 

0.126 

1.669 

0  222 

1.2 

0.182 

3.320 

0.521 

0.301 

9.479 

1.509 

0.179 

1.811 

0.258 

1.3 

0.262 

3.669 

0.565 

0.273 

9.435 

1.698 

0.230 

1.971 

0.295 

1.4 

0.336 

4.055 

0.608 

0.247 

9.392 

1.904 

0.280 

2.151 

0.333 

1.5 

0.405 

4.482 

0.651 

0.223 

9.349 

2.129 

0.328 

2.352 

0.372 

1.6 

0.470 

4.953 

0.695 

0.202 

9.305 

2.376 

0.376 

2.577 

0.411 

1.7 

0.531 

5.474 

0.738 

0.183 

9.262 

2.646 

0.423 

2.828 

0.452 

1.8 

0.588 

6.050 

0.782 

0.165 

9.218 

2.942 

0.469 

3.107 

0.492 

1.9 

0.642 

6.686 

0.825 

0.150 

9.175 

3.268 

0.514 

3.418 

0.534 

2.0 

0.693 

7.389 

0.869 

0.135 

9.131 

3.627 

0.560 

3.762 

0.575 

2.1 

0.742 

8.166 

0.912 

0.122 

9.088 

4.022 

0.604 

4.144 

0.617 

2.2 

0.788 

9.025 

0.955 

0.111 

9.045 

4.457 

0.649 

4.568 

0.660 

2.3 

0,833 

9.974 

0.999 

0.100 

9.001 

4.937 

0.690 

5.037 

0.702 

2.4 

0.875 

11.02 

1.023 

0.091 

8.958 

5.466 

0.738 

5.557 

0.745 

2.5 

0.916 

12.18 

1.086 

0.082 

8.914 

6.050 

0.782 

6.132 

0.788 

2.6 

0.956 

13.46 

.1.129 

0.074 

8.871 

6.695 

0.826 

6.769 

0.831 

2.7 

0.993 

14.88 

1.173 

0.067 

8.827 

7.406 

0.870 

7.473 

0.874 

2.8 

1.030 

16.44 

1.216 

0.061 

8.784 

8.192 

0.913 

8.253 

0.917 

2.9 

1.065 

18.17 

1.259 

0.055 

8.741 

9.060 

0.957 

9,115 

0.960 

3.0 

1.099 

20.09 

1.303 

0.050 

8.697 

10.018 

1.001 

10.068 

1.003 

3.5 

1.253 

33.12 

1.520 

0.030 

8.480 

16.543 

1.219 

16.573 

1.219 

4.0 

1.386 

54.60 

1.737 

0.018 

8.263 

27.290 

1.436 

27.308 

1.436 

4.5 

1.504 

90.02 

1.954 

0.011 

8.046 

45.003 

1.653 

45.014 

1.653 

5.0 

1.609 

148.4 

2.171 

0.007 

7.829 

74.203 

1.870 

74.210 

1.870 

6.0 

1.792 

403.4 

2.606 

0.002 

7.394 

201.7 

2.305 

201.7 

2.305 

7.0 

1.946 

1096.6 

3.040 

0.001 

6.960 

548.3 

2.739 

548.3 

2.739 

8.0 

2.079 

2981.0 

3.474 

0.000 

6.526 

1490.5 

3.173 

1490.5 

3.173 

9.0 

2.197 

8103.1 

3.909 

0.000 

6.091 

4051.5 

3.608 

4051.5 

3.608 

10.0 

2.303 

22026. 

4.343 

0.000 

5.657 

11013. 

4.041 

11013. 

4.041 

log,  X  =  (logio  x)^M ;    M=  .4342944819.        logio  6^+"  =  logm  e^  +  logjo  ev. 

Sinhx  and  coshx  approach  e^/2  as  x  increases  (see  Fig.  E,  p.  20).  The 
formula  logio  (e V2)  =  M  •  x  —  logio  2  represents  login  sinh  x  and  logio  cosh  x  to 
three  decimal  places  when  x  >3.5 ;  four  places  when  x  >  5 ;  to  five  places  when 
X  >  6 ;  to  eight  places  when  x  >  10. 


V,  E] 


ELLIPTIC   INTEGRALS 


53 


D.     VALUES   OF 
doc 


-'o  Vl-A:2sin-'e     -'O  v^ 


sine 
siii<}>, 


[Elliptic  Integral  of  the  First  Kind.] 


0.0 

.i.  =  o- 

<fr  =  10' 

,/.  =  15' 

./.  =  30' 

*=45» 

.fr  =  60» 

<i.=;6"' 

/i 

=71/36 

=ir/18 

=  ,r/12 

=  .r/6 

=  7r/4 

=  77/3 

=677/12 

*=90» 

=  77/2 

0.087 

0.175 

0.262 

0.524 

0.785 

1.047 

1.309 

1.571 

0.1 

0.087 

0.175 

0.262 

0.324 

0.78(> 

1.049 

1.312 

1.575 

0.2 

0.087 

0.175 

0  262 

0.325 

0.78!) 

1.054 

1.321 

1.588 

0.3 

0.087 

0.175 

0.262 

0.526 

0.792 

1.062 

1.336 

1.610 

0.4 

0.087 

0.175 

0.262 

0.527 

0.798 

1.074 

1.358 

1.643 

0.5 

0.087 

0.175 

0.26.3 

0.529 

0.804 

1.090 

1.385 

1.686 

0.6 

0.087 

0.175 

0.263 

0.5.32 

0.814 

1.112 

1.42() 

1.752 

0.7 

0.087 

0.175 

0.263 

0.5;«) 

0.82(5 

1.142 

1.488 

1.8.54 

0.8 

0.087 

0.175 

0.264 

0  539 

0.839 

1.178 

1.566 

1.9!  13 

O.i) 

0.087 

0.175 

0.264 

0.544 

0.8,58 

1.233 

1.703 

2.275 

10 

0.087 

0.175 

0.265 

0.549 

0.881 

1.317 

2.028 

00 

i;(/c,  <t>)=  pvi-fessiiiaerferr  r 


VALUES   OF 


dx. 


VI  -  ic2 

[Elliptic  Integral  of  the  Second  Kind.] 


[  x  =  sin  e  ' 
[t«  =  sin<J)^ 


fc= 

*  =  5° 

<i>  =  10'' 

.^=15'' 

*=30° 

*=45° 

«=60° 

</.  =  75° 

/; 

=  77/36 

=  77/18 

=77/12 

=77/6 

=77/4 

=77/3 

=577/12 

*=90'' 

=77/2 

0.0 

0  087 

0.175 

0.262 

0.524 

0.785 

1.047 

1.309 

1..571 

0.1 

0.087 

0.175 

0.262 

0.523 

0.785 

1.046 

1.306 

1.566 

0.2 

0.087 

0.174 

0.262 

0.523 

0.782 

1.041 

1.297 

1  .,5.54 

0.3 

0.087 

0.174 

0.262 

0.521 

0.779 

1.033 

1.283 

1.5.33 

0.4 

0.087 

0.174 

0.261 

0.520 

0.773 

1.026 

1.2f)4 

l.,504 

0.5 

0.087 

0.174 

0.261 

0.518 

0.767 

1.008 

1.240 

1.467 

0.6 

0.087 

0.174 

0.261 

0.515 

0.759 

0.989 

1.207 

1.417 

0.7 

0.087 

0.174 

0.2(;0 

0.512 

0.748 

0.<)65 

1.163 

1..351 

0.8 

0.087 

0.174 

0.2()0 

0.50!) 

0.7.37 

0.910 

1.117 

1.278 

0.9 

0.087 

0.174 

0.259 

0.505 

0.723 

0.907 

1.0,53 

1.173 

1.0 

0.087 

0.174 

0.259 

0.500 

0.707 

0.866 

0.966 

1.000 

54  NUMERICAL  TABLES 

F.     VALUES  OF  U(p)  =  T (p  +  1)  =  Ce- ^xPdx 
p  A  PROPER  FRACTION 

[n  (n)  =  r  («  +  1)  =  ?i!,  if  M  is  a  positive  integer.] 


[V 


/J=0.0 

p=0.1 

p=0.2 

2>^0.3 

1>=0.4 

i>=0.6 

i»=0.6 

jp=0.7 

p=0.8 

P= 

r(p+i)= 

1.000 

0.951 

0.918 

0.897 

0.887 

0.886=v^/2 

0.894 

0.909 

0.931 

0.9 

r(A;  +  1)  =  A;r(^),  if  A;>0;  lience  r(A;  +  l)  can  be  calculated  at  intervals  of  0.1 
Minimum  value  of  r(/>  +  1)  is  .885(30  alp  =  .46163. 


G.  VALUES  OF  THE  PROBABILITY  INTEGRAL: 


^^d. 


X 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

0. 
1. 
2. 

.0000 
.8127 
.9953 

.1125 

.8802 
.9970 

.2227 
.9103 
.9981 

.3286 
.9340 
.9989 

.4284 
.9523 
.9993 

.5205 
.9661 
.9996 

.6039 
.9763 
.9998 

.6778 
.9838 
.9999 

.7421 

.9891 
.9999 

.796 
.992 
l.000( 

H.     VALUES   OF  THE  INTEGRAL     f 

[Note  break  at  x  =  0.] 


^&^dx 

00    X 


n  =  l 

11=2 

n=^3 

»=4 

n=5 

/t=6 

»i  =  7 

n  =  8   n  = 

x=—n/10 

-  .2194 
— 1.823 

-  .0489 
-1.223 

-  .0130 

-  .9057 

-  .0038 

-  .7024 

-  .0012 

—  .5598 

-  .0004 

-  .4544 

-  .0001 

—  .3738 

-.0000  -.0( 
-  .3106  -  .2* 

.t=+n/10 
x=+n 

-1.623 

1.895 

-  .8218 
4.954 

-.3027 
9.934 

+  .1048 
19.63 

.4542 
40.18 

.7699 
85.99 

1.065 
191.5 

1.347   l.< 
440.4   10 

— -  =—  00.     Values  on  each  side  of  x  =  0  can  be  used  safely. 

(* '  -^  and   r  —  dx  reduce  to  the  integral  here  tabulated ;  see  IV,  99,  104,  p.' 
Jo  logx  J  x" 


V,I] 


RECIPROCALS      SQUARES      CUBES 


55 


RECIPROCALS    OF   NUMBERS    FROM    1    TO  9.9 


1 

.0 

.1 

.2 

3 

.4 

.5 

.6 

.7 

.8 

.9 

l.O(X) 

0.f)09 

0.833 

0.7G!) 

0.714 

0.fiG7 

0.625 

0.588 

0.556 

0.526 

2 

0.500 

0.47tJ 

0.455 

0.4.35 

0.417 

0.400 

0.385 

0.370 

0.357 

0.345 

3 

0.833 

0.323 

0.313 

0.303 

0.2M 

0.28(! 

0.278 

0.270 

0.263 

0256 

0.2r>0 

0.244 

0.238 

0.2.33 

0.227 

0.222 

0.217 

0.213 

0.208 

0.204 

0-,'00 

0.196 

0.192 

0.189 

0.185 

0.183 

0.179 

0.175 

0.172 

0.169 

0.1G7 

O.KA 

O.KJl 

0.151t 

0.15t) 

OArA 

0.152 

0.14it 

0.147 

0.145 

0.143 

0.141 

0.139 

0.137 

o.i;» 

0.133 

0.132 

O.VM 

0.128 

0.127 

0.125 

0.123 

0.122 

0.120 

0.119 

0.118 

0.116 

0.115 

0.114 

0.112 

9 

0.111 

0.110 

0.109 

0.108 

0.106 

0.105 

0.104 

0.103 

0.102 

0.101 

I2.     SQUARES    OF   NUMBERS   FROM    10   TO  99 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

100 

121 

144 

169 

196 

225 

256 

289 

324 

3(il 

2 

400 

441 

484 

529 

576 

625 

676 

729 

784 

841 

3 

iKX) 

i)61 

1024 

1089 

1156 

1225 

129() 

1369 

1444 

1521 

4 

1600 

1681 

1764 

1849 

1936 

2025 

2116 

2209 

2;»4 

2401 

5 

3500 

3601 

2704 

3809 

2916 

3025 

3136 

3249 

3364 

3481 

6 

.3(i(X) 

3721 

3844 

39(i9 

4096 

4225 

4^56 

4489 

4624 

4761 

7 

4ft00 

5041 

5184 

5329 

5476 

5625 

5776 

5929 

fW84 

6241 

8 

WOO 

(i561 

6724 

6889 

7056 

7225 

7396 

7569 

7744 

7921 

9 

8100 

8281 

84M 

8649 

88;i6 

9025 

9216 

9409 

9604 

9801 

I3.      CUBES    OF   NUMBERS   FROM    1    TO 


.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

1 

1.00 

1.33 

1.73 

2.20 

2.74 

3.37 

4.10 

4.91 

5.83 

6.86 

2 

8.00 

9.26 

10.65 

12.17 

13.82 

15.62 

17.58 

19.68 

21.95 

24.39 

3 

27.00 

29.79 

32.77 

35.94 

39.:J0 

42.87 

46.66 

50.65 

54.87 

56.32 

4 

64.0 

68.9 

74.1 

79.5 

85.2 

91.1 

i)7.3 

103.8 

110.6 

117.6 

5 

1350 

132.7 

140.6 

148.9 

157.5 

166.4 

176.6 

1852 

1951 

205.4 

6 

216.0 

227.0 

238.3 

250.0 

262.1 

274.6 

287.5 

.•100.8 

314.4 

328.5 

7 

;i43.0 

357.9 

373.2 

389.0 

405.2 

421.9 

4.39.0 

456.5 

474.6 

493.0 

8 

512.0 

5.31.4 

551.4 

571.8 

.592.7 

614.1 

6;i«).l 

658.5 

(W1.5 

705.0 

9 

729.0 

753.6 

778.7 

804.4 

s:«).(> 

8.-)7.4 

884.7 

912.7 

!m.2 

970.3 

56 


NUMERICAL  TABLES 


[V,J 


Ji.     SQUARE   ROOTS   OF    NUMBERS    FROM    1    TO   9.9 


.0 

.1 

.2 

.3 

.4 

.6 

.6 

.7 

.8          .9 

0 

0.000 

0.316 

0.447 

0.548 

0.632 

0.707 

0.775 

0.837 

0.894    0.949 

1 

1.000 

1.049 

1.095 

1.140 

1.183 

1.225 

1.265 

1.304 

1.342    1.378 

2 

1.414 

1.449 

1.483 

1.517 

1.549 

1.581 

1.612 

1.643 

1.673    1.703 

3 

1.732 

1.761 

1.789 

1.817 

1.844 

1.871 

1.897 

1.924 

1.949    1.975 

4 

2.000 

2.025 

2.049 

2.074 

2.098 

2.121 

2.145  • 

2.168 

2.191    2.214 

5 

2.336 

2.?58 

3.380 

3.303 

3.334 

3.345 

3.366 

3.387 

3.408     2429 

6 

2.449 

2.470 

2.490 

2.510 

2.530 

2.550 

2.569 

2.588 

2.608    2.627 

7 

2.046 

2.665 

2.683 

2.702 

2.720 

2.739 

2.757 

2.775 

2.793    2.811 

8 

2.828 

2.846 

2.864 

2.881 

2.898 

2.915 

2.933 

2.950 

2.966    2.983 

9 

3.000 

3.017 

3.033 

3.050 

3.066 

3.082 

3.098 

3.114 

3.130    3.146 

Jo.     SQUARE   ROOTS   OF   NUMBERS   FROM   10  TO 


0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

1 

3.162 

3.317 

3.464 

3.606 

3.742 

3.873 

4.000 

4.123 

4.243 

4.359 

2 

4.472 

4.583 

4.690 

4.796 

4.899 

5.000 

5.0il9 

5.196 

5.292 

5.385 

3 

5.477 

5.!i68 

5.657 

5.745 

5.831 

5.916 

6.000 

6.083 

6.164 

6.245 

4 

6.325 

(>.403 

6.481 

6.557 

6.633 

6.708 

6.782 

6.856 

6.928 

7.000 

6 

7.071 

7.141 

7.311 

7.380 

7.348 

7.416 

7.483 

7.550 

7.616 

7.681 

a 

7.746 

7.810 

7.874 

7.937 

8.000 

8.062 

8.124 

8.185 

8.246 

8.307 

7 

8.367 

8.426 

8.485 

8.544 

8.602 

8.()60 

8.718 

8.775 

8.832 

8.888 

8 

8.944 

9.000 

9.055 

9.110 

9.165 

9.220 

9.274 

9.327 

9.381 

9.434 

9 

9.487 

9.539 

9.592 

9.644 

9.695 

9.747 

9.798 

9.849 

9.899 

9.950 

K.     RADIANS   TO   DEGREES 


Radians 

Tenths 

HUNDRKDTIIS 

Thousandths 

Ten-thousandths 

1 

57°17'44".8 

5°43'46".5 

0°34'22".6 

0°  3'26".3 

0°  0'20".6 

2 

114°35'29".6 

11°27'33".0 

1°  8'45".3 

0°  6'52".5 

0°  0'41".3 

3 

171°53'14".4 

17°11'19".4 

1°43'07".9 

0°10'18".8 

0°  1'01".9 

4 

229°10'59".2 

22°55'05".9 

2°17'30".6 

0°13'45".l 

0°  1'22".5 

5 

28(i''28'44".0 

28°38'52".4 

2°51'53".2 

0°17'11".3 

0°  l'43".l 

6 

343°46'28".8 

34°22'38".9 

3°26'15".9 

0°20'37".6 

0°  2'03".8 

7 

401°  4' 13"  .6 

40°  6'25".4 

4°  0'.38".5 

0°24'03".9 

0°  2'24".4 

8 

458°21'.58".4 

45°.50'11".8 

4°35'01".2 

0°27'30".l 

0°  2'45".0 

9 

515°39'43".3 

51°33'58".3 

5°  9'23".8 

0°30'56".4 

0°  3'05".6 

V,  M] 


CONSTANTS 


IMPORTANT  CONSTANTS  AND  THEIR  COMMON 
LOGARITHMS 


.V=N,-M1!KK 

Vaiie  of  .V 

Lo.:,„  .V 

IT 

3.14109265 

0.49714987 

1-f-  JT 

0.31830989 

9.5028.501.3 

ir2 

9.86i)60440 

0.9942997.-. 

VF 

1.77345385 

0.24S.57494 

e  =  Napierian  Base 

2.71828183 

0.434J944S 

M  =  logiQ  e 

0.43429448 

9.(;;i77.S4.'.l 

1-^1/=  log,  10 

2.30258509 

0.3(i221.-)(i'.) 

LSO  -r-  TT  =  degrees  in  1  radian 

57.2957795 

1.75812262 

IT  -^  180  =  radians  in  1° 

0.01745329 

8.24187738 

7r  H-  10800  =  radians  in  1' 

0.0002fl08882 

6.4637261 

TT  --  (U8000  =  radians  in  1" 

0.0000O4S4813(!811095 

4.68557487 

sin  1" 

0.0000048481.Tj81107r) 

4.68557487 

tan  1" 

0.00000484813tJ811152 

4.(58557487 

centimeters  in  1  ft. 

30.480 

1.48401.58 

feet  in  1  cm. 

0.032808 

8.5159.'v42 

inches  in  1  m. 

39.37 

1.5951654 

pounds  in  1  kg. 

2.204G2 

0.3433;i40 

kilograms  in  1  lb. 

0.453593 

9.656(>660 

g 

32.16  ft./sec./sec. 

1.5073 

=  981  cm. /sec. /sec. 

2.9916690 

weight  of  1  cu.  ft.  of  water 

62.425  lb.  (max.  density) 

1.7953+ 

weight  of  1  cu.  ft.  of  air 

0.0807  1b.  (at32°F.) 

8.907 

cu.  in.  in  1  (U.  S.)  gallon 

231 

2  3636120 

ft.  lb.  per  sec.  in  1  H.  P. 

550. 

2.7403627 

kg.  m.  per  sec.  in  1  H.  P. 

76.0404 

1.8810445 

watts  in  1  H.  P. 

74.-..957 

2.8727135 

DEGREES  TO  RADIANS 


r 

.01745 

10° 

.17453 

100° 

1.74533 

6' 

.00175 

6" 

..H.,0. 

.o:M91 

20" 

..34907 

110° 

1 .91986 

7' 

.00204 

7" 

.00003 

.3° 

.052:36 

.-30° 

.52360 

120° 

2.09440 

8' 

.00233 

8" 

.00004 

4^ 

.06981 

40° 

.69813 

V.W 

2.26893 

9' 

.00262 

9" 

.ax)04 

5° 

.08727 

.50° 

.87266 

140° 

2.44.'H(J 

10' 

.00291 

10" 

.00005 

6'^ 

.10472 

(«" 

1.04720 

1,50° 

2.61799 

20' 

.00582 

20" 

.(MM)1() 

7'^ 

.12217 

7()0 

1.22173 

1(»° 

2.79253 

m' 

.00873 

.30" 

.(KM)15 

S'^ 

.13<N)3 

«0° 

l.;i!l62() 

170° 

2.9670<5 

40' 

.OHM 

40" 

.(MMM9 

9^^ 

.1.5708 

itO'^ 

1  ..570.H0 

180° 

3.141.59 

.50' 

.014.54 

.50" 

.IH)(I'_'4 

58 


NUMERICAL  TABLES 


[V,  N 


N.     SHORT    CONVERSION    TABLES    AND   OTHER    DATA  : 
MULTIPLES,  POWERS,  ETC.,  FOR  VARIOUS  NUMBERS 


n  =  l 

M  =  2 

w=8 

n=4 

n  =  6 

n  =  6 

n=7 

n=8     n=9 

TT  .71 

3.1416 

6.2832 

9.4248 

12.566 

15.708 

18.850 

21.991 

25.133-  28.274 

W  .  7(2/4 

.78540 

3.1416 

7.0686 

12.566 

19.6.35 

28.274'38.485 

50.265    63.617 

TT .  n3/6 

.523()0 

4.1888 

14.137 

33.510 

65.450 

113.10 

179.59 

268.08    381.70 

TT-f-  n 

3.14] 6 

1.5708 

1.0472 

.78540 

.62382 

.52360 

.44880 

.39270    .34907 

n-~  T 

.31831 

.63662 

.95493 

1.2732 

1.5915 

1.9099 

2.2282 

2.5465    2.8648 

(TT/ISO)  .  n 

.01745 

.03491 

.05236 

.06981 

.08727 

.10472 

.12217 

.13963    .15708 

(180/7r)  .  n 

57.296 

114.59 

17189 

229.18 

286.48 

343.77 

401.07 

458.37   515.66 

e  -71 

2.7183 

5.4366 

8.1548 

10.873 

13.591 

16.310 

19.028 

21.746    24.465 

M-n 

.43429 

.86859 

1.3028 

1.7371 

2.1714 

2.6057 

3.0400 

3.4744   3.9087 

(1  ~M)-n 

2..3026 

4.6052 

6.9078 

9.2103 

11.513 

13.816  16.118 

18.421    20.723 

\~n 

1.0000 

.50000 

.33333 

.25000 

.20000 

.16667 

.14286 

.12500    .11111 

ri2 

1. 

4. 

9. 

16. 

25. 

36. 

49. 

64.         81. 

7l3 

1. 

8. 

27. 

84. 

125. 

216. 

343. 

512.       729. 

n* 

1. 

16. 

81. 

256. 

625. 

1296. 

2401. 

4096.     6561. 

ns 

1. 

32. 

243. 

1024. 

3125. 

7776. 

16807. 

32768.    59049. 

25.2" 

64. 

128. 

256. 

512. 

1024. 

2048. 

4096. 

8192.     16384. 

3" 

3. 

9. 

27. 

81. 

243. 

729. 

2187. 

6561.     19083. 

V^Tt 

1. 

1.4142 

1.7321 

2. 

2.2361 

2.4495 

2.6458 

2.8284       3. 

v'« 

1. 

1.2599 

1.4422 

1.5874 

1.7100 

1.8171 

1.9129 

2.       2.0801 

n\ 

1. 

2. 

6. 

24. 

120. 

720. 

5040. 

40320.   862S30. 

l-=-n! 

1. 

0.5 

.16667 

.04167 

.00833 

.00139 

.00020 

.00002    .000003 

£«* 

1^6 

1. 

l-fSO 

5. 

l-f42 

61. 

1^30 

1385.    5-f66 

cm.  in  n  in. 

2.5400 

5.0800 

7.6200 

10.160 

12.700 

15.240J  17.780 

20.320    22.860 

in.  in  n  cm. 

.39370 

.78740 

1.1811 

1.5748 

1.9685 

2.3622 

2.7559 

3.1496   ;3.5433t 

m.  in  n  ft. 

.30480 

.60960 

.91440 

1.2192 

1.5240 

1.8288 

2.1336 

2.4384    2.7432 

ft.  in  n  m. 

3.2808 

6.5617 

9.8425 

13.123 

16.404 

19.685 

22.966 

26.247    29.527 

Itm.  in  n.  mi. 

1.6093 

3.2187 

4.8280 

6.4374 

8.0467 

9.6561 

11.265 

12.875    14.484 

mi.  in  n  km. 

0.6214 

1.2427 

1.8641 

2.4855 

3.1069 

3.72824.3496 

4.9710   5.5923 

kg.  in  ?t  lb. 

.45359 

.90719 

1.3608 

1.8144 

2.2680 

2.72163.1751 

3.6287    4.0823 

lb.  in  n  kg. 

2.2046 

4.4092 

6.6139 

8.8185 

11.023 

13.22815.432 

17.637    19.842 

1.  in  n  qt. 

.946.36 

1.8927 

2.8391 

3.7854 

4.7318 

5.67826.6245 

7.5709   8.5172 

qt.  in  n  1. 

1.0567 

2.1134 

3.17004.2267 

5.2834 

6.3401i7.3968 

1 

8.4534   9.5101 

*  £„  =  7ith  Bernoulli  number;  see  II,  E,  15-18,  p.  8. 
\  Exact  legal  values  in  U.  S. 


INDEX 


[Numbers  in  roman  type  refer  to  pages  of  the  body  of  the  book;  those  in 
italics  refer  to  pages  of  the  Tables.] 


Absolute  value,  16. 

Acceleration,  71  ;  angular,  81  ;  com- 
ponent, 74 ;  of  a  reaction,  90 ; 
relative,  156;  tangential,  71  ;  total, 
74. 

Adiabatic  expansion,  293. 

Algebraic  functions,  45. 

Amplitude,  of  S.  H.  M.,  157. 

Analytic  geometry,  formulas,  14- 
See  also  Curves. 

Anchor  ring,  11. 

Annuity,  5. 

Approximate  integration,  239,  43- 

Approximation.  See  also  Error,  La- 
grange, Prismoid,  Simpson,  Tay- 
lor. 

Approximations,  formulas  for,  4^! 
polynomial,  227,  250,  28;  Simp- 
son-Lagrange, 29;  Taylor,  257, 
273,   28;   trigonometric,  8,  29. 

Arbitrary  constants,  346. 

Area,  polar  coordinates,  212 ;  of  a 
surface,  334  ;  surface  of  revolution, 
129,  336. 

Areas,  103,  46. 

Astroid,  24- 

Asymptotes,  264,  268,  311. 

Atmospheric  pressure,  143. 

Attraction,  226,  48. 

Auxiliary  equation,  differential  equa- 
tion, 366. 

Average,  of  section  area,  126. 

Average  pressure,  249. 

Average  value,  217,  220,  47. 

Bacterial  growth,  143. 
Beams,  79,  209,  296,  374. 
Bernoulli  numbers,  58. 
Bessel'a  functions,  280. 


Binomial  differentials,  195,  39. 
Binomial  theorem,  276,  7. 

Cardioid,  24. 

Cassinian  ovals,  27. 

Catenary,  139,  140,  20. 

Cavalieri's  Theorem,  125. 

Center  of  gravity,  217,  219,  220,  47. 

Center  of  mass,  47-     See  also  Center 

of  gravity. 
Centroid.     iSee  Center  of  gravity. 
Chance,  6. 
Circle,  9,  15. 
Circular  measure,  19,  151.     See  also 

Radian. 
Cissoid,  26. 

Clairaut  equation,  362. 
Coefficient  of  expansion,  27,  145. 
Combinations,  6. 

Compound  harmonic  functions,  158. 
Compound    interest    law,    141,    168, 

230. 
Conchoid,  26. 
Cone,  318,  11. 
Confocal  quadrics,  343. 
Constants,  1,  58. 

Continuity.     See  Function,  continu- 
ous. 
Contour  lines,  294,  319,  25. 
Conversion  tables,  58. 
Cooling,  in  fluid,  142. 
Critical  point,  on  a  surface,  322. 
Critical  values,  for  extremes,  63. 
Cubes,  table  of,  55. 
Curvature,  169,  305;    center  of,  171, 

305 ;   circle  of,  305 ;  radius  of.  170, 

305. 
Curves,     17,     see     also     Functions; 

cubic,   S5;    parabolic,   16,   17,   see 


59 


60 


INDEX 


also     Polynomials;     quartic      '^o  ■ 
in  space,  319.  '    ~    ' 

Curve  tracing,  313. 

Curvilinear  coordinates,  331 

Cusp,  310. 

Cycloid,  155,  307,  311   22 

Cylinder,  319,  10;  projecting,  331. 


Double     integrals,     210.     See     also 

Integrals. 
Double  Law  of  the  Mean,  267. 
Double  point,  310. 


Damping,    of    vibrations,    160,    368 

Damping  factors,  163. 
Definite  integrals,  44. 
Depreciation,  5. 

Derivative,     22;      directional,     294- 
of  a  constant,  29 ;    of    a  function 
of  a  function,  36 ;  of  a  power,  29, 
33,   39;    of    a    product,    35;    of    a 
quotient,  33  ;   of  a  sum,  29  ;    loga- 
rithmic, see  Logarithmic ;    partial 
281,    16;   total,  285. 
Derivatives,   notation    for,    23;   sec- 
ond, 71;    of  inverse  trigonometric 
functions     163;     of    exponentials, 
139  ;    of  logarithms,  134  ;    trigono- 
metric functions,  150. 
Derived  curves,  77,  241. 
Determinant,  5. 
Difference  quotient,  6. 
Differential,  partial,  288 ;   total,  286 
360.  *  ' 

Differential  coefficient,  23. 
Differential  equations,  94,  159  I60 
345;  extended  linear,  358;  higher 
order,  375  ;  homogeneous,  354  ;  lin- 
ear, 356;  linear,  constant  coeffi- 
cients, 375 ;  non-homogeneous,  369, 
^77;  ordinary,  348;  partial,  284, 
382  ;  second  order,  363,  366 ;  sepa- 
rable, 353  ;  systems  of,  379. 
)ifferential  formulas,  52, 173, 15    See 

also  Derivatives. 
)ifferentials,     50;      complete,     344 
notation   for,    50;    transformation 
01,  16. 

'irection  cosines,  315. 
'irectional  derivative,  294. 
istribution  of  data,  28. 


Electric  current,  146,  284. 

Elimination  of  constants  348 

Ellipse,  9,  15. 

Ellipsoid,  318,  11,  14. 

Elliptic  functions,  5S. 

Elliptic  integrals,     ^ee  Integrals. 

Empirical  curves,  227. 

Energy  integral,  364,  373. 

Envelopes,  298. 

Epicycloid,  23. 
Epitrochoid,  22. 

Equations,  differential,  see  Differen- 
tial;   in   parameter   form,   47,  see 
also  Parameter ;   solution  of,  44  4 
Error,   240,   254,    288,   see    also    Ap- 
proximation ;    curve,  28. 
Errors,  of  observation,  48. 
Evolute,  172,  300. 
Explicit  functions,  45. 
Exponentials,      138,     20,      see      also 
Logarithms ;       differentiation     of 
139  ;   table  of,  52. 
Exponents,  3. 
Extremes,  8,  63,  260,  322;   final  tests 

for,  64,  75,  325  ;  weak,  323. 


Factors,  4. 

Falling  bodies,  100,  206. 

Family,  of  curves,  35. 

Finite     differences,     251.     See     also 

Increments. 
Flexion,  71. 
Flow  of  heat,  294. 
Flow  of  water,  280,  295. 
Fluid  pressure.     See  Water  pressure. 

Atmospheric  pressure,  etc. 
Folium,  46,  48,  63,  26. 
Force,  82  ;  work  done  by,  249  48 
Fourier's  Theorem,  8,  29. 
Fresnel's  integrals,  279. 
Frustum,  of  a  cone,  11 ;  formula,  122  • 

of  a  solid,  121. 
Functions,  1 ;     continuous,    14      17  ■ 
derived,  23;    notation  for,   3;   im- 
phcit,    etc.,    see   Implicit,  etc.;    of 


INDEX 


61 


functions,  36;  algebraic,  rational, 
etc.,  see  Algebraic  functions,  etc. ; 
classification  :o:f^  28. 


>n,  271,  54. 

>n  of,  43,  57,  90,  137, 


Gamma'T 
Gases,  ex' 

142,  292r^^^ 
Geometry,  spkce,  315 
Graphs,  2. 
Gudermannian,  166,  279,  13. 
Guldin  and  Pappus,  Theorem,  47. 
Gyration,  radius  of.     See  Radius. 

Harmonic  functions,  21.  See  also 
Trigonometric. 

Helicoid,  332. 

Helix,  332. 

Hooke's  Law,  157. 

Hyperbola,  10,  15. 

Hyperbolic  functions,  139,  140,  13, 
20,  52;  inverse,  see  Inverse. 

Hyperbolic  logarithm.  See  Napier- 
ian. 

Hyperboloid,  318,  31. 

Hypocycloid,  24. 

Hypotrochoid,  23. 

Implicit  functions,  45,  294. 

Improper  integrals,  201.  .See  also 
Integrals. 

Increments,  6,  252 ;  method  of,  233  ; 
second,  234,  see  also  Finite  differ- 
ences. 

Indeterminate  forms,  263,  268. 

Inertia,  moment  of.     See  Moment. 

Infinite  series.     See  Series. 

Infinitesimal,  17  ;  principal  part,  266. 

Infinitesimals,  higher  order,  265. 

Infinity,  19. 

Inflexion,  point  of,  75. 
ntegral,  as  limit  of  sum,  116  ;  funda- 
mental theorem,  99  ;  indefinite,  96  ; 
notation  for,  96. 

Integral  curves,  241,  350,  380. 

Integrals,  definite,  100,  44;  double, 
210;  elliptic,  195,  247,  280,  9,  53; 
Fresnel's,  279;  improper,  201, 
314;  multiple,  217;  table  of,  33; 
triple,  217. 


Integral  surfaces,  of  a  differential 
equation,  380. 

Integrand,  96. 

Integraph,  244. 

Integrating  factor,  361. 

Integration,  96;  approximate,  239, 
see  also  Approximation  ;  by  parts, 
181,  33;  by  substitution,  176, 
33,  41 ;  formulas  for,  97,  174,  33; 
of  a  sum,  97 ;  of  binomial  differen- 
tials, 195,  39;  of  irrational  func- 
tions, 164,  37;  of  linear  radicals, 
188,  37  ;  of  polynomials,  97,  176 ; 
of  quadratic  radicals,  189,  37; 
of  rational  functions,  184,  34; 
of  trigonometric  functions,  178, 
188,  194,  200,  39;  reduction  formu- 
las, 196,  39,  40;  repeated,  206; 
successive,  206. 

Interpolation,  Lagrange's  formula, 
15.     See  also  Lagrange. 

Inverse  functions,  46,  3. 

Inverse  hyperbolic  functions,  166, 
247.  14,  52. 

Inverse  problems,  347.  ,See  also 
Rates,  reversed. 

Inverse  trigonometric  functions,  163. 

Involute,  301. 

Irrational  functions,  29 ;  differen- 
tiation of,  38;  integration  of,  164 
189. 

Isolated  point,  310. 

Isothermal  expansion,  137,  292. 

Lagrange  interpolation  formula,  228, 

15,  45. 
Laplace's  equation,  284,  344. 
Law  of  the  Mean,  251,  45;   double, 

267;       extended,     257,     see      also 

Taylor's  theorem. 
Least  squares,  69,  229,  262,  323,  342, 

6. 
Length,  18,  106,  46;  of  a  space  curve, 

338. 
Lemniscate,  27. 

Limit    {  l+M".  271. 


(-1)" 


Limits,  16  ;  arc  to  chord,  18 ;  proper- 
ties of,  17  ;  sin  a  to  a,  19. 
Liquid  pressure,  48. 


62 


INDEX 


Loci,  318.     See  also  Curve  tracing. 

Logarithmic  derivative,  146,  167. 
See  also  Rates,  relative. 

Logarithmic  plotting,  229,  18. 

Logarithms,  computation  of,  7  ; 
graph  of,  19;  hyperbolic,  see 
Logarithms,  Napierian ;  modulus 
M,  134;  Napierian,  135,  53; 
natural,  see  Napierian;  rules  of 
operation,  130,  3;  table  of,  50. 

Maclaurin's  Theorem,  258.     See  also 

Taylor's  Theorem. 
Mass,  47. 

Maximum,  8.     See  also  Extremes. 
Mean  square  ordinate,  248. 
Mensuration,  9. 

Minimum,  8.     »See  also  Extremes. 
Modulus,  of  logarithms,  134. 
Moment,  first,  224. 
Moment   of  inertia,   213,   47;    polar 

coordinates,  214. 
Momentum,  82. 
Motion,     107,     363,     48.     See     also 

Speed,  Acceleration,  etc. 

Napierian     base,    e,    135.     See    also 

Logarithms. 
Natural  logarithms.    See  Logarithms. 
Normal,  11,  59;    length  of,  60;    to  a 

surface,  327,  330. 
Notation,  1. 
Numbers,  e,  M.     See  Logarithms. 

Organic  growth,  law  of,  143. 
Orthogonal  trajectories,  361. 

Pappus'  Theorem,  47. 

Parabola,  10.  See  also  Curves,  para- 
bolic. 

Paraboloid,  11,  32. 

Parameter  forms,  47,  59,  107,  332. 

Partial  derivative,  281,  see  also 
Derivative  ;    order  of,  283. 

Partial  derivatives,  transformation, 
339,  16. 

Partial  differential.     See  Differential. 

Partial  fractions,  184. 

Pendulum,  254,  280. 


Percentage    rate    of    increase,    144. 

<S'ee  also  Rates. 
Period,  of  S.  H.  M.,  157. 
Permutations,  6. 
Phase,  of  S.H.M.,  157. 
Plane,  equation  of,  316. 
Planimeter,  245. 
Point  of  inflexion,  75. 
Polar  coordinates,  5,  166  ;  plane  area, 

212;      moment    of    inertia,     214; 

space,  332. 
Polynomial,     approximations.       <See 

Approximations. 
Polynomials,    28,    see    also    Curves, 

parabolic  ;    differentiation  of,   29  ; 

roots  of,  38,  57  ;  integration  of,  97, 

176. 
Power  curves,  17. 
Power  series.     See  Taylor  series. 
Primitive,  of  a  differential  equation, 

349. 
Prism,  10. 

Prismoid,  definition  of,  126. 
Prismoid  rule,  125,  239,  10,  45. 
Probability,  6,  see  also  Least  Squares  ; 

Error;    curve,    28;    integral,    280, 

48,  54. 
Pyramid,  10. 
Pythagorean  formula,  107. 

Quadric  surfaces,  318,  31;   confocal, 

.343. 
Quartic  curves,  25. 

Radian  measure,  table  of,  49,  56. 

Radium,  dissipation  of,  146. 

Radius  of  curvature.  See  Curva- 
ture. 

Radius  of  gyration,  216,  219,  48. 

Rates,  6,  23,  49;  average,  22; 
instantaneous,  23 ;  percentage, 
144;  related,  85;  relative  144, 
146,  167,  353  ;  reversal  of,  91,  345, 
see  also  Integrals;  time,  12,  70. 

Rational  functions,  28 ;  difTerentia- 
tion  of,  32  ;  integration  of,  184  . 

Rationalization,  of  radicals.  See  In- 
tegration. 

Reactions,  rates  of,  90,  143,  146. 

Reciprocals,  table  of,  55. 


INDEX 


63 


Reduction  formulas,  196,  39. 

Relative  rate  of  increase,  144,  146, 
167.  See  also  Rates  and  Loga- 
rithmic derivative. 

RoHp's  Theorem,  250. 

Roulettes,  22. 

Series,  alternating,  277 ;  convergence 
tests,  272  ;  differentiation  of,  278  ; 
gpomotric,  272,  7 ;  infinite,  271,  7 ; 
integration  of,  278 ;  precautions, 
276  ;   Taylor,  see  Taylor. 

Simple  harmonic  motion,  155,  .365,  21. 

Simpson-Lagrange  approximations, 
29. 

Simpson's  rule,  129,  240,  45. 

Singular  point,  309. 

Singular  solution  of  a  differential 
equation,  362. 

Slope,  6. 

Solution  of  equations,  4- 

Specific  heat,  27. 

Speed,  12,  71,  see  also  Motion ; 
component,  14 ;  angular,  81  to- 
tal, 49,  74,  107  ;   of  a  reaction,  90. 

Sphere,  318,  11. 

Spirals,  30. 

Square  roots,  table  of,  56. 

Squares,  table  of,  55. 

Strophoid,  25. 

Subnormal,  60. 

Subtangent,  60. 

Summation,  approximate,  111,  see 
also  Approximations  and  Integral ; 
exact,  114,  see  also  Integral; 
step  by  step,  110. 

Summation  formula,  115,  211. 

Surfaces,  quadric,  318,  31. 


Table  of  integrals.  196,  33. 

Tables.     See  special  titles. 

Tangent,  equation  of,  7,  58;  length 
of,  60  ;  to  a  space  curve,  337. 

Tangent  plane,  to  a  surface,  321,  329. 

Taylor  series,  273,  28. 

Taylor's  Theorem,  257,  8,  see  also 
Law  of  the  Mean;  several  vari- 
ables, 341. 

Time  rates.     iSee  Rates. 

Total  derivative,  285.  iSee  also 
Derivative. 

Total  differentials.  See  Differen- 
tials. 

Tractrix,  24. 

Trajectories,  orthogonal,  361. 

Transcendental  functions,  29,  130. 

Trapezoid  rule,  239,  4o. 

Trigonometric  functions,  150;  table 
of,  12,  19,  49. 

Trigonometry,  9. 

Trochoid,  22. 

Variable,  1 ;  dependent,  2  ;  indepen- 
dent, 2. 

Velocity,  71.     See  aZso  Speed. 

Vibration,  157,  365,  21;  damped, 
160,368,22;  electric,  157. 

Volume,  of  frustum,  121 ;  solid  of 
revolution,  123. 

Volumes,  120,  212,  46. 

Water  pressure,  117,  48. 
Waves,  157. 
Witch,  26. 

Work,  of  a  force,  249,  48;  on  a  gas, 
138,  142,  48. 


T 


HE    following  pages    contain   advertisements  of  a 
few  of  the   Macmillan  books  on  kindred  subjects. 


AN  ELEMENTARY  TREATISE  ON  THE  CALCULUS 

With  illustrations  from  Geometry,  Mechanics,  and  Physics.  B) 
GEORGE  A.  GIBSON,  Professor  of  Mathematics  in  the  Glasgow 
and  West  of  Scotland  Technical  College.  i2mo.  Cloth,  xix  + 
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The  author  has  written  a  book  primarily  for  the  students  of  mathemat- 
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In  the  early  chapters  the  theory  of  graphs  and  of  units  is  discussed, 
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Another  innovation  is  made  by  the  introduction  of  a  chapter  on  the  Theory 
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INTRODUCTION  TO  THE  CALCULUS 

Based  on  Graphical  Methods.  By  GEORGE  A.  GIBSON,  Professor 
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A  PRIMER  OF  CALCULUS 

By  ARTHUR  S.  HATHAWAY,  Professor  of  Mathematics  in  the 
Rose  Polytechnic  Institute,  Terra  Haute,  Indiana.  i2mo.  Cloth 
viii  +  139  pages,     fi.25  7iet. 

BRIEF  INTRODUCTION  TO  THE  INFINITESIMAL  CALCULUS 

By  IRVING  FISHER,  Professor  of  Political  Economy  in  Yale 
University.     i2mo.     Cloth.     vii+ 84  pages.     75  cents  ;/^/. 

Thii  little  book  is  the  result  of  a  desire  to  help  readers  without  special 
mathematical  training  to  understand  mathematical  and  economic  works. 
It  is  equally  adapted  to  the  use  of  those  who  wish  a  short  course  in  the 
calculus  as  a  matter  of  general  education.  Teachers  of  mathematics  will 
find  it  useful  as  a  text-book  in  courses  planned  for  the  general  student. 

INTEGRAL  CALCULUS  FOR  BEGINNERS 

With  an  introduction  to  the  study  of  Differential  Equations.  By 
JOSEPH  EDWARDS.    i2mo.  Cloth.    xiii  + 308  pages.    $1.10;/^/. 

AN   ELEMENTARY   TREATISE   ON   THE   DIFFERENTIAL 
CALCULUS 

With  Applications  and  Numerous  Examples.  By  JOSEPH 
EDWARDS.     8vo.     Cloth.     xiii  + 529  pages.     $3.50  «^^. 

DIFFERENTIAL  CALCULUS  FOR  BEGINNERS 

By  JOSEPH  EDWARDS.    i2mo.    Cloth.    x  + 262  pages.    $1.10  net. 
A  TREATISE  ON  THE  DIFFERENTIAL  CALCULUS 

With  Numerous  Examples.  By  I.  TODHUNTER.  i2mo.  Clolh. 
viii  +  420  pages.     $2.60  net. 

TREATISE    ON    THE    INTEGRAL    CALCULUS    AND    ITS 
APPLICATIONS 

With  Numerous  Examples.  By  I.  TODHUNTER.  l2mo.  Cloth. 
vi  +  480  pages.     $2.60  fiet. 


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COLLEGE  MATHEMATICS 

ALGEBRA 

Introduction  to   Higher  Algebra     By  Maxime  Bocher, 

Professor  of  Mathematics  in   Harvard   University 

Cloth        8vo        3'5  pcge^        $1.90  net 

A  text  for  those  students  who  will  take  up  the  study  of  higher  mathe- 
matics. It  fills  the  gap  between  college  algebra  as  ordinarily  taught  and 
the  subjects  taken  in  higher  mathematics.  The  author  has  undertaken 
to  introduce  the  student  to  higher  algebra  in  such  a  way  that  he  shall 
become  familiar  with  both  the  proofs  of  the  most  fundamental  algebraic 
facts  and  with  the  many  important  results  of  algebra  which  are  new  to 
him. 

Advanced   Algebra      By  Arthur  Schultze,  Ph.D.,  Assistant 

Professor  of  Mathematics,   New  York  University  ;    Head  of 

the    Mathematical   Department,    High   School   of  Commerce, 

New   York    City  Half  leather        s(>^  P^g^^        $1.25  net. 

Designed  particularly  with  the  view  to  give  the  student  such  a  working 
knowledge  of  algebra  as  will  ordinarily  be  required  in  practical  work. 
Graphical  methods  are  emphasized  more  than  is  general  in  books  of  this 
grade.  This  will  be  found  an  excellent  text  for  all  students  desiring 
technical  knowledge.  The  book  is  furnished  with  or  without  answers 
as  desired.  The  edition  without  answers  will  be  sent  if  no  choice  is 
indicated. 

Graphic   Algebra      By  Arthur  Schultze 

Cloth         J 2 mo         93  pfiges         $  .80  net 

This  book  gives  an  elementary  presentation  of  the  fundamental  principles 
included  in  the  courses  usually  given  on  this  subject  and  contains,  in 
addition,  a  number  of  methods  which  are  shorter  and  require  less 
numerical  work. 

A   Treatise   on   Algebra      By  Charles  Smith,   M.A. 

Cloth         646  pages         $1.90  net 

An  excellent  text  for  classes  in  college  which  deal  with  the  subject  largely 
from  the  theoretical  sifle;  nevertheless,  care  has  been  taken  to  get  a 
large  number  of  problems  illustrative  of  the  principles  demonstrated. 
Answers  to  the  problems  are  included. 


COLLEGE  MATHEMATICS  -  Continued 


COORDINATE   GEOMETRY 

Analytic  Geometry  for  Technical  Schools  and  G)l- 

leges       By  p.    A.    Lambert,   Assistant  Professor  of  Mathe- 
matics,  Lehigh  University 

Cloth         i2mo         216  pages         $1 .50  net 

The  object  is  to  furnish  a  natural  but  thorou£;h  introduction  to  the  prin- 
ciples and  applications  of  analytical  geon.etry  with  a  view  to  the  use 
made  of  the  subject  by  engineers.  The  important  engineering  curves  are 
thoroughly  discussed  and  the  applicat'  m  of  analytic  geometry  to  mathe- 
matics and  physics  is  made  a  special  ^joint. 

Conic   Sections       By  Charl'.s  Smith,   Master  of  Sidney  Sussex 

College,  Cambridge       Cw/k        127,10        352  P^ges        $1.60  net 

This  well-known  text,  which  has  been  reprinted  eighteen  times  since  the 
second  edition  was  issued  in  1883,  is  still  considered  a  standard.  The 
elementary  properties  of  the  straight  lines,  circle,  quadrille,  ellipse,  and 
hyperbola  are  discussed,  accompanied  by  many  examples  selected  and 
arranged  to  illustrate  principles. 


CALCULUS   AND   DIFFERENTIAL  EQUATIONS 

A  First  Course  in  the  Differential  and  Integral  Cal- 
culus By  William  F.  Osgood,  Professor  of  Mathematics 
in  Harvard  University  ^^^^^        ^^^        ^^.00  net 

Designed  as  a  text  for  students  beginning  the  study  and  devoting  to  it 
about  one  year's  work.  The  principal  features  of  the  book  are  the  in- 
troduction of  the  integral  calculus  at  an  early  date;  the  introduction  of 
the  practical  applications  of  the  subject  in  the  first  chapters;  and  the  in- 
troduction of  many  practical  problems  of  engineering,  physics,  and 
geometry.  The  problems,  over  900  in  number,  have  been  chosen  with  a 
view  to  presenting  the  applications  of  the  subject  not  only  to  geometry, 
but  also  to  th«  practical  problems  of  physics  and  engineering. 


COLLEGE  ALGEBRA 

BY 

SCHUYLER  C.   DAVISSON,  Sc.D. 

Professor  of  Mathematics  in  Indiana  University 


Cloth,  i2mo,  ig I  pages,  $1.50  net 


A  discussion  of  those  parts  of  algebra  usually  treated  in  the 
first  year's  course  in  college.  The  author  aims  that  the  student 
shall  acquire  not  merely  a  comprehension  of  algebraic  processes, 
but  the  ability  to  use  without  difficulty  the  language  of  algebra  — 
to  express  in  his  own  language  conclusions  ordinarily  expressed  in 
symbolic  form,  and  thus  gain  the  ability  to  generalize  easily. 

A  characteristic  feature,  developed  in  the  course  of  several 
years  of  teaching  college  freshmen,  is  the  introduction  early  in  the 
course  of  the  fundamental  laws  of  algebra.  When  the  student 
once  recognizes  these  foundations  and  the  continuity  of  the  sub- 
ject is  pointed  out  to  him,  there  will  be  a  higher  degree  of  interest 
in  the  facts  of  algebra  accompanying  a  more  intelligent  compre- 
hension of  their  relations,  and,  in  consequence,  they  will  be  more 
readily  retained  and  more  easily  applied  in  later  work. 


PUBLISHED    BY 

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